I didn't use Polo for many years, so I cannot really tell from experience, but $2k is a rather small amount and they should not cause troubles, especially as their help page tells:
If anything is centralized, they are centralized. Anything centralized and that has control on your money can ask for KYC at anytime, even if you want to withdraw the lowest amount that is withdrawable on the exchange.
Kucoin is a centralized exchange but it allows trading and withdrawing when you don't want KYC, but you can only withdraw up to 1BTC within 24 hours with a non-KYC account.
I have not used Kucoin before, but thwre are sometimes the exchange request for KYC whike you want to withdraw any amount, be it small amount or large amount. I registered on OKX some months ago after Binance make KYC mandatory, later OKX asked me for phone number verification. I verified my phone number before I was able to withdraw.