How does one conquer Native Americans and not loose diplomatic points?

The only way to avoid your situation would have been to not fully annex the Aztecs. This is largely undocumented, but fully annexing an overlord will transfer all of their vassals to you. Next time, you get in such a situation, you will be able to make a better assessment. If you take on vassals, you must consider, that each will take a diplomatic relation and that each relation over your limit costs 1 diplomatic point per month.

The main mitigating option you have is to sever other diplomatic relations (alliances and so on). Or try to get in a war you lose and offer to release your vassals as tribute. The former isloates you in europe, the latter isn't guaranteed to work, as your enemy may not care about vassals in central america.