First, your distinction between "normal people", which are moral, and "rich people", who are just evil, are kind of disgusting. Having a lot of money doesn't automatically make one evil (to remove all suspicion, I am not rich and likely never will be, at least by US standards :). I'm not sure what your point is there, but it doesn't sound good.
> the most effective way to do that would have been to campaign against certain presidential candidates
That's certainly not the most effective way. US law has very clear immigration provisions, which require certain documentation, and US president swears to uphold the law. Most efficient strategy would be to change the law, thus voting for specific - namely, open-border - candidate for Congress, that can change the law so that US has immigration procedures no longer, and thus does not have to deport people who violate them. Note that I did not say "efficient" but "most efficient" - I see no really efficient strategy that would make US a country without immigration law anytime soon, and having such law, it is natural to have to enforce it - otherwise there's no point in having it.
In any case, nothing prevents you to from campaigning for or against any policy or any candidate, be it open border or not, as a private person or as a part of an organized effort. Why you need the church to do it? Just create an independent organization and campaign as your hear pleases.
> I'm obviously not inclined to say "Please start taxing my church,"
If you register a non-profit and make all the requirements that tax-exempt non-profits have, and do charitable work and so on, nothing prevents you from calling your organization "church", but I don't see why doing so should give you any special status not available to any other nonprofit. And if it doesn't qualify as nonprofit, then it should be taxed by the same rules as the rest of organizations are.