Today we finally diagnosed the backfiring situation on my volvo penta 5.7. It all started last week when we removed the water separator and emptied it out and replaced without filling it with gas. So we turn the boat on and it does exactly what yours did, idles great but when you give it some throttle it would start backfiring like it had bad gas but that was impossible since I filled it with Rec 90 non-ethanol fuel. Turns out the fix was very easy in my case maybe some of what I did might help you.
First make sure the engine is cold if you're going to do this and you turn the blower on to air out the engine compartment. All we did was without the engine turning over we kept turning the key until the pump kicked on and on the pump itself there is a schrader valve like the one on your tire. We pushed it in and it bled some gas and air a couple of times but after a couple of times clicking the key on and off it started shooting straight gas. Then we took the plastic cover on top of the engine off and toward the back of the throttle body there is another schrader valve and we did exactly the same procedure and it shot some air and some gas and finally only gas. We took the boat to the lake today and it ran like a bat out of hell!

I hope this will help you out since I know exactly how annoying that engine backfiring can be. Good luck getting your situation taken care of. Keep us posted
