I've never heard of Binance taking action against transactions within their exchange and gambling site.

I doubt that was the case but mostly, accounts locks are due to some other reasons like flagged and alarm transactions, unauthorized login attempts, or something along those lines, etc. Binance users are free to withdraw up to BTC2 per day without KYC so I conclude that they are not that strict too on gambling transactions. However, if the amount involved is decent, just withdraw it in batches.

Using a non-custodial wallet is the recommended one in terms of "Safety" but if you are regularly funding your gambling site account, you can directly transact with Binance.

Or if you really want to make sure that everything goes without a problem then;

- Deposit fiat and purchase crypto at your preferred exchange
- Transfer to your non-custodial wallet
- Transfer to the gambling site

For withdrawals

- Gambling site > Non-Custodial Wallet > Exchange