Wow, that is a lot of root. I don't have a great deal of experience growing, but I do have two thoughts. One is that judging from the size of what your picture shows above ground, many of my plants may be similarly root bound, but they seem to be happy in the pots there in. I'm curious what prompted you to check on whether they outgrown the 5 gallon pots because maybe my happy plants would be even happier with more room.
Second, I have heard that some plants like to be "underpotted" (though not undernurished, of course). I don't understand the logic there. Swaddling a baby makes sense, but I don't get why a plant would be positively effected by a little squeeze. I can see how the ecosystem for a container garden would be different than a more traditional - in the ground - garden. How big of an issue is being root bound for the health of pepper plants?