I don't know if my unverified Kucoin account is just special, but it shows a 24-hour withdrawal limit of 5 BTC (currently ~$200,000)
Yes, kucoin has 5 btc withdrawal limit/day instead of 2 btc, it was before IIRC.
I have used binance over a year, no kyc, and without having a problem so I guess you can trade your $5k there, its up to you though.