I knew a personal shopper who used her own American Express for all purchases (get those points/miles/etc), and then just billed the customer via ACH afterward. She would invoice via QuickBooks (Free ACH payments), but there are programmable options out there.

Plaid+Stripe can hook into a bank account, and you can initiate an ACH that way. The ACH fee is only 0.8% with a cap at $5. That might be low enough to eat it. I'm not sure what your business model is though.