Under the Spanish colonial system, Chile and La Plata were colonies governed separately and of different hierarchical rank .. while one was a viceroyalty the other (Chile) was a general captaincy dependent and ruled by the viceroyalty of Peru.
Chile on the other hand was liberated by an army formed in and from La Plata (which was joined by the remnants of the defeated revolutionary/independentist Chilean forces). After achieving the victory and therefore the situation of fact of the Platense army was the unique military force organized there and by the patriotic solidarity (common to the time and region) ... for a short period of time Chile, in fact, depended and was integrated into the Plata.
But the hypothetical small 'window of opportunity' (for the objectives required for the OP) of a permanent integration into a larger political organization was closed and/or wad a los /'failed' opportunity in OTL mostly by the centralist / unitary ideology and monarchical policies that from Platense revolutionary leadership supported from part and the consequent political and military chaos provoked by the rebellions and regional political challenges to these policies, the ambiguity of the political-economic objectives of the 'Porteña' (from Bs. As.) oligarchy and the subsequents repression/civil wars.
But in any case and given the above described (synthetically) political situation that made impossible in OTL, any desire or possibility of union lost attractive and viability for the Chileans while reinforcing their logical autonomist / independentistas tendencies and finally the same question of a Possible membership / political union with the Río de la Plata, become superfluous after the 'Liberating Army' departed north to attack the realistic bastion of the Viceroyalty of Peru.
Therefore, in my opinion, the necessary conditions, but not enough, for a more or less permanent political union to take place would be, on the one hand, that the offensive against Peru would not be continued, while the liberation army remained in Chile.
On the other hand and more important a 'fast' and permanent triumph of the federalist / autonomist political / military forces.
This would at the same time avoid the institutional chaos and political-military anarchy (with logical material destruction) of OTL, with the logical advantages derived from the fact that they would probably establish the type of government than being attractive to the Chileans and other peripheral proto nationalities).
Could offer Chileans a respectful integration and which would guarantee their administrative and political autonomy while giving them the economic and political advantages of belonging to a common nation / federation.
A Supra Nation of that type formed by the majority of the territories of the former Viceroyalty of the Plata and Chile ...
it's probably that the war against Peru would continue ... but with the 'goal' to reconquer the territories of the 'Upper Peru' (that were formed, approximately, by part of extreme north Argentino and present-day Bolivia) and eventually continue the offensive against the Peruvian realists, from there rather than invading from the Pacific coast as it should be done, in OTL).
Finally, relations with Brazil would be tense at best, as long as it remains as a Monarchy and it is more likely that sooner or later there will be an open conflict to determine the hegemony of the region (for the possession / control of waterways and Its hinterlands) and / or by the real or perceived support, by the Emperor of Brazil, of the probable Brazilian secessionist / republican movements.
Finally about the Falklands, their importance was more for the whale's hunters ships refuge and eventually a resupply place for them and/or the Royal Navy (and one place that anyway wasn't OTL, very important or strategic for British.
But it's possible that the Falklands/Malvinas islands TTL, would have been more populated than in OTL and with a strong state with a stable central government a suitable garrison would have been sent/ established and there would be at least some ships destined there.
Also would be regular communications / transportation with the mainland instead of the isolation of OTL.
Its importance could have been increased for the hypothetical new Bioceanica nation of the 'southern cone'.
Because these islands would have become, probably on a scale, more or less obligatory / necessary for the naval traffic and of course for the maritime trade both from Chile towards the Plata and vice versa.