So what is the alternative? I coinjoined first time, but I haven't sent to any CEX yet, I'm thinking of sending these coins to some of my other wallets before (in my possession).
I am not sure. Staying away from centralized exchanges that can't be trusted is good advice regardless if you use privacy tools or not. Moving the coinjoined funds around within your non-custodial wallets like normal money isn't bad either. If a CEX freezes your funds, you can always say you bought the bitcoin from someone P2P and you have no way of knowing what they used them for in the past. That is if you don't want to tell your CEX you used a coinjoin service.
Honestly, it doesn't cost anything for a CEX to ask for depositing proofs claiming to be in compliance with current tax regulations. I say this because I had a similar experience, of course it wasn't due to the use of coinjoin, but rather failing to properly store my password + 2FA code.. In short, during my account recovery process, one of the requested items was proof of fiat deposits (I had made one in BTC).
But you're right, the way is to transfer coinjoined UTXOs to other wallets in my possession, wait a while before sending them to a custody service, but the more we can avoid CEX, the better!
Like using bisq. However, from time to time I need to use CEX due to investments such as derivative futures.