I think fee's is how exchanges make their money, so I think it would be unlikely that's an option. Happy to be educated otherwise though.

I use Binance where the Faster Transfer to deposit and withdraw fiat is free, but to transfer to another wallet (e.g. a Ledger or whatever) costs about £15. I was advised it may be best to work out your acceptable loss and then transfer when you hit that minimising your transfer fees.

I'm doing similar to you but £50 a month. I stick £50 on Binance and then watch the rate and do Limit buys of what I'm prepared to pay.