Well, I'm Swiss Catholic, but I guess I'll try my best...
Honestly, the biggest impact of Zwingli surviving the Battle of Kappel, assuming he and his faction of cantons get their way, isn't even theological: it's political and military.
It's severely understated just how much of the Kappelkrieg and by extension the Swiss Reformation and Counter-Reformation was related to a cultural debate over the role of Swiss mercenaries. Zwingli in particular, being something of a "nationalist" (if you can really apply that concept at this point in history), railed against the mercenary system as a waste of young Swiss men's lives fighting for foreign wars that should have been of no concern to the Confederation. IIRC, there was also some moralistic justification, that going to war in foreign countries was something of a doorway to sin, but don't quote me on that. This was a message that had fairly strong support among the urban cantons that had more diversified economies, where the Reformation took root most of all, while the poorer Catholic cantons that depended on the system rejected that idea in particular, and that usually coincided with remaining Catholic and embracing the Counter-Reformation in opposition to Zwingli. It's not a one-to-one correlation, but it was certainly a tendency that's pretty easy to see in which cantons sided with the Reformation and which stayed behind the Church.
Zwingli gets free reign? You see a drastic downturn in Swiss mercenaries and a more inward-looking Confederation, at least for a time. That will have a whole host of effects on Europe beyond the Alps and also probably means Switzerland is probably relatively poorer. On the other hand, if Zwingli and his allies were able to decisively achieve their aims, you have a decent recipe for an early centralization movement in politics and culture. Sure, Bern and Zurich will still not want the other involved in their business and maintain some level of loose confederation between them, but the especially reticent rural and Catholic cantons would probably be under some form of military or puppet rule and forced to abide by the larger cantons' wishes more closely than in real history, much like Aargau was under Bern for most of Swiss history.
With no PoDs prior to 1531, how different can the Reformation in Switzerland in the short term? What I’m specifically wondering here is, can the theology of Double Predestination and Election be significantly reduced in influence during the 16th Century?
Suppose, for example, that Zwingli wasn’t killed in the Battle of Kappel? Would it, for example, be enough for John Calvin to be slightly less successful in Geneva (even if he still writes his 1536 book)? Or is Calvin’s role in pushing predestination overstated? And if something like this is possible, what are the reverberations on the Reformation over the next century?
As mentioned above, while I'm sure Zwingli definitely believed his interpretation of the Bible should spread beyond Switzerland's borders, he was ultimately a pretty Swiss-focused figure. His greatest praise for Luther was more for his independence and challenge against the Catholic Church, rather than his theological arguments, as an example. In addition, he notably made many writings advocating for a joining of Church and State, recognizing a "godly" City Council of Zurich as a legitimate engine of the word of God and saw aristocracy as the preferred form of government in comparison to monarchy and democracy. He's a uniquely Swiss figure that would probably not translate as well to wider contexts in that era, so I speculate you would probably see someone else in another region rise to the role of Calvin as a wider unity figure within that particular sect of Christianity.
With regards to Calvin, I'll say that as far as I can understand, Zwingli (or his successors) also held to Double Predestination, even before the Helvetic Confession established accords between the Calvinist and Zwinglist Churches in Switzerland. On the other hand, keep in mind that Calvin was actually kicked out of Geneva at least once, IIRC,
after the city's conversion to Protestantism. Zwingli living might indeed neuter Calvin's specific program in the city, as a unitary figure for the Swiss Protestant Churches to hold to might have strengthened the resistance to Calvin's strain of Reformed Protestantism. That could cause its own ripples within the movement. In addition, the Helvetic Confession established by Bullinger and Calvin's followers was something of a blueprint to establish unity with other Reformed traditions, so absent that, you might have a more fractured and thus less popular or more localized set of Reformed Protestant Christian denominations.
I'm not sure about doctrine, but how about this? If Zwingli lives beyond 1531 the Bible is translated, either by him or under his influence, into a Swiss-German dialect, most probably that of Zurich. The Reformed Churches in other cantons adopt this Bible in preference to Luther's, and tracts and other religious works are written in the language. Gradually canton by canton adopts this as the language of administration, even, eventually the Catholic cantons out of practical necessity. Finally by the 'Age of Enlightenment' secular writers are using it to write poems and novels, so that by say, 1750 Switzerland has its own 'national' language, 'Schwyzerisch', as different from standard High German as Dutch is.
While I like the idea, I'll quickly point out that Schwiizertuutsch dialects are already mutually unintelligible from High German, with the latter being referred to as Schriifduutsch or "Writing German" in Switzerland itself. What you're proposing isn't the development of a language, merely a formalization and standardization of the common tongue. The only reason Swiss German isn't considered its own language in OTL is because it's got no standard written form and so doesn't have enough prestige to break out from the dialect title, though there is at least some literature in various local traditions that could have formed the basis of a Swiss German literary tradition.
To add to your POD here, by 1531, the Swiss are effectively independent from the Holy Roman Empire. A formalized language creates another early wedge of identity between the Swiss national identity and that of the wider German one, plus one that might extend into the other neighboring regions with a High Alemannic dialects like parts of Baden and Wurtemburg. Theoretically, it would also have knock-on effects in relations with the other linguistic regions within the Confederation. That could be positive, in which the local Swiss dialects of French and Italian are formalized, as I would love to see formalizations of Ticinese and Rhaet-Romansch as well (and if nothing else, maybe the damn French could use Septante, Huitante, and Nonante too due to Swiss influence instead of the absurd constructions of modern Standardized French for those numbers). On the other hand, tying the Swiss identity to a formalized Swiss German identity, as opposed to just saying "Everybody's from backwater mountain peoples and none of us really speak 'proper', no point getting upset about other languages in other cantons," could spell a cultural division that might lead to more ethnic or sectarian violence in the Confederation and potentially a smaller Switzerland.