You can, and we did.
We had an accounting software that was used in some pretty big Martial Arts schools, we handled their payments through Authorize.Net (stripe didn't exist yet), but in the contract, we told them a reversal will cost (making the numbers up because it has been so long) $25 + the transaction and a bounced eCheck was $30 + transaction. These fees were directly pass through, we didn't try to profit on them.
That said, the number of fraudulent card/check or bounces was 0.
We were the payment processor for a business, not working with skeezy customers directly (like Duet is).
If there is fraud, you bill that back to YOUR customer. Just like Stripe bills you for a bounced check or a reversal, you bill your SaaS customer.