I have done a bit more research and I'm thinking about avoiding Monero and mixers as it can give me problems with proof of source of funds later. Does anybody know if this is the same if you use atomic swaps but with other coins?
As far as I know you can technically always prove the source of funds if you keep all your transaction data, it doesn't matter if you use Monero or Lightning. If you use mixers, this may be different as there is a centralized intermediary. While they often provide a letter of guarantee signed by them which contains the deposit address and withdrawal address, an authority could distrust the certificate if they don't have a list of mixers.
Does the atomic swap only work for Monero or does it also work with, for instance, USDT? That is, could I use atomic swap for BTC -> USDT -> BTC and still succeed to un-KYC my coins? Would this approach still create problems for future proof of source of funds?
Atomic swaps with adaptor signatures and other private techniques exist for several coins, but I don't know if there's a software solution to exchange directly to ERC20 tokens like USDT (or on another blockchain). But for your purpose, you should also be fine if you exchange BTC to ETH and then back to BTC.
This approach for example works with Ethereum.
Take into account that it's common to re-use addresses in Ethereum, but you should do like you would do with Bitcoin-style coins and realize some transactions in ETH with different addresses ("accounts") before you change it back to BTC to maximize privacy.
To my mind, it's not a good idea to use Grin because first of all, its CMC is 2.5 million (nothing) and daily trading volume is very low, a few thousand dollars. Also, compared to Monero, it's harder to exchange Grin.
Yes, it's perhaps not ideal for beginners. However it's one of the few serious Monero competitors so I mentioned it.
MimbleWimble - the Grin approach to privacy - can also used with Litecoin through the MWEB function.
You have to make multiple transactions at different times to make it impossible to get tracked.
Exactly, this is very important in all techniques mentioned, I mentioned it also in one of my earlier posts.