Could a small tetrapod fly using Neopteran-like wings that are integrated into the ribcage between the ribs and the spine, with the ribcage being flexible and containing flight muscles instead of lungs
The main problem here would be the ribcage being used for flight. The main issue here is that insect wings are essentially extensions of their exoskeleton with muscles attached at the base, while the ribcage in itself is a fixated group of bones meant to protect internal organs. While in insects the wings are no biggie, in this scenario we'd be experiencing a very fast flattening of the body due to the ribcage going to the sides, which is actually a strategy somewhat used by flying snakes (flatten the body by expanding the vertebrae, allowing for some level of gliding). But in your case, if it was possible to begin with, you'd be putting such a strain on the internal organs by the constant flattening and returning and stretching of the skin that I doubt the creature would be able to fly for long or live to fly a second time.
So yeah, the use of a flapping ribcage on a vertebrate for flight sounds like something you wouldn't see happening. But, with the correct structure and some fusing of the ribs, so long as the shoulder blades didn't get in the way, you could likely create a structure capable of sustaining both the wings and the muscles necessary for their flapping. Now all you'll need is a something as effective, if not more, as the avian respiratory system, because insect flight requires decent level of oxygenation going directly to their muscles, so your lungs will need to make up for that.
A number of reptiles, past and present, have evolved the ability to glide using "wings" which were not attached to either pair of limbs. Wikipedia mentions Draco (extant), Xianglong (both lizards), Mecistotrachelos, the kuehneosaurids and weigeltisaurids. Many of these creatures look quite impressive.
Powered flight is another question, and I don't have enough background to comment beyond saying that gliding seems to have arisen many more times than powered flight among vertebrates and there is presumably a reason for this.
Since you've already made the comparison to insects and specified that the lungs are not contained in the ribs, can I assume the lungs are instead in the abdomen? In that case, then the answer is certainly yes. Others have already given examples of lizards with wings evolved from ribcages, and powered flight has evolved numerous times besides. In fact, if the ribs just connected to muscles attached to the arms, it could have powered flight. There's not really a reason to trade in the lungs for muscles, especially since flying animals need a great deal of oxygen (birds are largely lung, extending even into their bones)! Even if you're specifying very rapid flight, that's already what hummingbirds do.
It's also worth noting that insect flight muscles are extremely simple. You have muscles to squish the thorax up, and muscles to squish it down, and that's what makes the wings move. Vertebrate wings are much more complex. I'm not sure why you want so much muscle space.