The lightning network was designed for cases like this... hundreds of microtransactions between 2 parties could result in only 2 onchain transactions (one opening and one closing the channel). The transactions are off-chain, instant, and nearly free (if you get your users to open a channel directly to your lightning node, you can even make sure their lightning transactions are completely free).

As for your problem, i don't see a real technical sollution... Whatever you do will always require some level of trust. Either you trust your users ( which is a bad idear imho), or your users trust you. I'd personally prefer the second case: you can build trust over a longer period of time, whilst your user's are unknown, faceless individuals whose accounts can even be sold to scammers.

There is a non-technical sollution tough: make your game provably fair, sign every deposit address and escrow a couple hundred (in fiat equivalent) with a trusted forum escrow.
If you ever scam your users, they should be able to provide screenshots of their bets, the seed, the hashes, the signed messages and turn to the escrow provider...
As soon as you've ran your business and processed several BTC, you can eliminate the escrow and just provide signed messages from the hot wallets proving you have processes a lot of BTC and *should* be trustable.