To update, I spoke with the fingerprint department in Middletown. They were able to verify in their system that they had in fact rejected my prints twice (matching Shelton's story).
The woman I spoke with also told me in no uncertain terms that people in this thread are misinforming me (the Darien guy, and the guy a few posts ago) who said they were able to have prints electronically verified by the state in minutes. She says the process is NEVER that fast regardless of the print method, and that "somebody is misinforming" me. Clearly somebody is misinformed, and it's probably not the people in this topic (they'd have no rational reason to troll the forum by making things up like that).
She said if I get a copy of the rejection letter from Shelton, I can take that up to Middletown and they could print me (at a cost), approve the prints, and send them back to Shelton so that the rest of the process could proceed. She said if I could NOT get that rejection letter, if I got fingerprinted in Middletown, they'd have to mail the prints to Shelton, then have Shelton turn around and mail them right back to Middletown for approval. She said the reason for this is that Shelton has all my permit information, and only the prints go to Middletown, and nothing else like background check or anything proceeds until after a "complete" application including accepted fingerprints exists.
This is turning into a massive headache. It sounds like whether I go to Middletown or Shelton, I'm in for a possibly lengthy wait yet again. Not sure what I want to do at this point. They have me close to the point of questioning whether this enormous hassle is even worth the effort. And that's probably exactly what they want ("they" being whomever put into place such a complicated process), to discourage applicants.
As for the side argument, I absolutely think it's absurd for prints to be required for this. They can tell right off the bat that I'm not a felon, and don't have a criminal record of any kind. I passed a background check a few years ago while buying a long gun. That didn't require fingerprints. I "submitted" to the prints because it's the only way to get a handgun permit in the state, rightfully or wrongfully. I highly doubt I could "appeal" the need for prints with any success.