There are a couple of solutions, all the payments can be done from the same address if you use a Mixing service, just put the payment address on the mixer output and that should do the magic.
Another option is to use another exchange with low fees or a casino to make the payments as a site withdraw.
Mixer service would slow down the transfer and would be quite expensive, right?
What other exchange do you recommend for this?
Certainly I wouldn't keep funds in a casino, lol.