Problem with that is the fact, that Sicilian is not that much more similar to Venetian than to Spanish.
Indeed.
What is specifically needed, to sum up several contributors:
1) A particular region--ideally one political realm, with a widespread single dialect or set of very closely related dialects all similar to the ruling political dialect--of Italy;
2) Sends large numbers of colonists, enough to get a major urban center and also populate the surrounding hinterland;
3) to a particular zone (or we can scrape up the numbers of settlers, zones, ideally adjacent to each other) where both urban center and hinterland are included;
4) without larger numbers of immigrants from other sources diluting them too much--how much is too much depends on how close the additional sourced immigrants are in language to the particular Italian dialect in point 1--neighboring Italians (or people we don't classify as Italians, if the core dialect is one considered more peripheral--say Venetian might IIRC have more in common with Dalmatian than Tuscan perhaps) can come in greater numbers than more linguistically distant Italians and those in greater numbers than more closely related Romance language speakers, those in greater numbers than distant Romance languages, versus Germanic, Slavic, Celtic, Greek, non-Indo European, etc.
But then again, if the number of persons coming from distant Italian sources and beyond are very heterogenous among themselves, so there is no big wave of Spanish speakers let us say, just a fair number trickling in over time at the same time a great diversity of others come in, then the metropolitan/colonial government Italian dialect would assimilate them all piecemeal. Unless of course a non-Italian or distant Italian dialect speaking power conquers management of the colony, so
5) political continuity of the founding Italian power must be maintained until either the colony is so large its native born population outnumbers a generation or two of immigrants, or until local secessionists who adhere to the founding dialect among themselves, mostly, take over and maintain control long enough to cross that boundary under their own independent management.
With these stringent conditions, we have to go astray pretty far ATL I think. First of all an independent Italian great power that is not a component of some empire under non-Italian management is unlikely; Italy was all entangled with the Holy Roman Empire and the Papal states and tended to come under German, French, or Spanish hegemony--or if we go back far enough or go with a stronger longer lasting "Rhomanian" TL, Greek.
(Harumph. I ought to say "Roman" there, and don't think there is any linguistic or historic excuse to spell it "Rhoman," never mind they are at war with the folks actually claiming the city of Rome in their territory, they still think of themselves as Romans full stop, never mind they speak Greek and look to Constantinople for rule. We put the h in there for one of the several reasons previous generations of west European scholars said "Byzantine" or the Latin contemporaries said "Greeks," because unless the actual city of Rome gets the Carthage treatment with no comeback, it is always confusing to talk about two antithetical sets of Romans! But the East Romans had no such confusion, they were the real Romans, whether Rome the city was in their control or not).
The way I read the OP, it is not necessary at all that the dialect called "Italian" for the challenge actually be the Florentine Tuscan that that official language of OTL is grounded on. It could be any dialect in the lands we call Italian today--Genoan or something else on the Ligurian coast, Neopolitan, Sicilian (Sardinian might be a stretch but hey, it's legally in modern OTL Italy, is it not?), Venetian--any of these distinct speeches or others one might name can qualify, if a power that makes this dialect rather than Florentine the official language and meets the other conditions well enough. Or Florentine of course, if Florence itself manages to secure some large and lasting secure bailiwick with enough population where the various grassroots dialects are close enough to Florentine to merge neatly enough. I scanted inland powers like Milan or Turin--or Rome!--there because I gather the inland lingo is different enough from any major port regional dialect to present problems.
Of course, if we have a whole diverse set of different dialects but they have a fair amount in common and one of them is strongly favored as the official language so that diverse-dialect immigrants find that splitting the difference among themselves is fairly similar to adopting the central ruling one, that might be enough of a melting pot--especially if don't mind a distinctly American version of the nominal official tongue! Perhaps one strongly influenced by adoption of native terms and African ones too; the relationship might be one like the strong Cockney influence on diverse English-speaking but lower class immigrants in Australia, New Zealand and South Africa making a distinct and common (more or less) "Colonial-English" from "the Queen's."
Anyway, either a single Italian power, which can be based anywhere, north, south, east or center we can make the case for it, must emerge and persist for centuries while maintaining control of its colony, or else we provide for a province within a larger empire that the masters of the large empire devolve control of their designated colony or colonies to and let them stick to it, while maintaining both the unity of the Italian province and its allegiance to the greater power, and withal the greater empire defending its hold on the delegated Italian colony or colonies.
The disadvantage of making the Italian metropolis a dependent part of a larger empire is obviously that the bigger power might quite likely find cause to reassign control of the colony to some other faction within its leading circles, particularly as an Italian fiefdom with internal unity and security is likely to want to run off on its own hook sooner or later and get out of step with its patron--possibly doing so successfully but then unable to maintain communication with its colony nor defend it from the betrayed patron power--if successful in so stabbing its patron in the back that the patron empire disintegrates, the colony is still more likely prey to, or even the paid price of help from, one of the great power rivals the Italian statelet connived with or faces unprotected. Or without any question of the Italian domain going out of its patron's grip, we might have the powers that be within the Empire just rationalizing control of the colony under other ethnic hands, presumably the dominant one.
But the advantage is to solve another problem Italian metropoli have that the Spanish, Portuguese, French, Dutch, British, and even the Scandinavian also-rans (except Sweden) don't have--access to the Atlantic. If we postulate an Italian-led west Med hegemony where the Italian power holds the Strait of Gibraltar firmly itself (via exarchates in Gibraltar and on the African opposite coast too) or is firmly allied to say a South Iberian power or Portugal or Spain, or some hypothetical Christianized Morocco or an unholy Catholic-Muslim alliance with Muslim northwest Africa, then the gate is open. But while I do think making say Genoa, its landward defenses augmented by having its back to the Alps, queen of the Western Med, holding all the islands and either an alliance or direct holding of south Iberia and the northwest cape of Morocco (or just one of these) would be fun, it might not seem very probable. More likely if such a hegemony were based on a much broader empire, if it say included Catalonia and/or Marseilles and as much else of south France's Med as one cares to name, perhaps a vast Provence, or were a checkerboard alliance of greater Portugal versus east Iberian and south French power (under Parisian rule or otherwise). It would be a heck of a lot more secure landward if say Savoy were an integral part of it, dynastically or constitutionally speaking; if it held the Po valley (and that opens the floodgates against the exclusively Genoan and close neighbor dialects background) and so on. And most rationally such a hodgepodge empire of diverse domains would integrate its colonies under one rule and whichever part of the patchwork of nations with the highest surplus population for emigration would tend to dominate the language.
We might have more fun not meeting the OP challenge but making a new creole Romance language from an eclectic mix of Italian dialects, Provencal, Catalonian, Baleric, Andalusian and Portuguese for instance, developing in opposition to a regime that uses Church Latin in writing and whose rulers shift back and forth between speaking Ligurian, Florentine, Provencal and Catalonian--or maybe while one or several of these dominate the royal court (or republican directorate, whatever) back in the west Med, the fact that Portugal is most convenient to provide the administrators makes it dominant instead in America.
So overall, I suspect the path of least resistance is:
1) Spain keeps hegemony over the OTL Spanish grants but
2) early on establishes an Italian stronghold they deepen their grip on--say they find themselves welcomed as rulers of Sicily and various policy choices make Sicily more populous and a strong source of emigration as well as a fraternal kingdom deemed a near-equal partner in the Spanish ruling hegemony, Or heck, Genoa or Naples--the former is easier to see getting rich enough to have surplus population, the latter is more extensive but hard to justify having more emigration and suitable unity and loyalty.
3) Early in Spanish colonial ventures, this Italian smaller twin of the dual core realm is subfranchised some piece of the Empire of the Indies, fully trusted to develop it in harmony with Spanish policy, and succeeds, drawing most of its European rulers from the Italian domain, while imposing their dialect on the Native people and any slaves brought in later, and maintaining a combination of settler growth and ongoing emigration from the Italian domain to keep pace with immigrants from other sources. Say the Italians don't have the "Castilian Peninsulares only can trusted to rule!" and eke out their trusted class of colonial rulers with those the Spanish would call "Criollos," that is, persons of Spanish (Castilian in fact, the Spanish were leery of letting non-Castilian Spanish subjects emigrate) descent--these faced hard limits on upward mobility in OTL Empire of the Indies, but perhaps their Italian-ancestry counterparts might not--say the Italian rulers did reserve the top spots for persons from either Italy or Castile, but distinguished among their "creoles" born in America, with individuals who have served with honor and distinction at low levels finding all but the highest levels open to them, and perhaps promoting the occasional person of mixed Native or African heritage (with more stringent scrutiny of course) on merit too, maybe someday going so far as to appoint an American-born person (presumably one who has gone very far to demonstrate strong ties to the Italian homeland, such as serving for decades in forces in Europe) supreme intendant of the Italian colony. This greater ability to leverage American born supporters might offset lower population potential which might enable say Sicily to run a Nova Sicily somewhere--I certainly am thinking, somewhere on the OTL "Spanish Main," that is northern South America. Or maybe a set of islands in the Lesser Antilles, producing a homegrown and reliable Caribbean based navy in support of the larger empire's security and logistics?