I think a lot of people, including the OP, are putting far FAR too much weight on 'but we've asked twice, and STILL no invitation was forthcoming!'.
The OP said that etiquette was a bit foreign to this side of the family. It also sounds like they are just super casual. Add in that the hosts are sleep-deprived parents of an infant. With all that information, let's try to see this from the hosts' possible perspectives. Also, do note that at some point "the story" was that the OP had in fact been sent a direct invitation that had been lost in the mail.
Possible Scenario #1: Host sends an invitation to his great uncle that is *intended* to include his daughter. Host assumes (wrongly) that everyone will know this. At some point host's father calls and says "Hey, did you mean to invite great-uncle's daughter?". Host replies "Of course! She's part of great-uncle's invite, duh!". Host believes their work is done.
If at that point, when the host feels they've been quite clear that the daughter was intended to be invited, the daughter were *still* to insist she hadn't actually been invited, host might reasonably feel like she was making a mountain out of a molehill and wonder why this was causing so much angst.
Possible Scenario #2: Host actually mails invitation directly to dad's cousin. It gets lost in the mail. At some point Host's father calls to ask if the daughter is invited. Host replies "of course she is, we mailed an invitation. If she didn't get it, it must have been lost in the mail. Make sure she knows." Host believes their work is done.
If at that point, when the host has actually invited the father's cousin, AND has sent communication through the family channels that an actual, physical, for real invitation was mailed to her, she were to still insist that since she'd never *received* an invitation that she wasn't *really* invited, I think it's quite likely that the host would be reasonably annoyed.
A few weeks before my cousin's wedding, my brother indicated that he had not received an invitation. I texted my cousin's fiancee to check. Her immediate response was that of course he was invited, she had sent his invitation out at the same time as everyone else's including a plus one. I relayed the information to my brother, who from that point forward operated as if he had received a physical invitation. If he had insisted that he wasn't *really* invited until either my cousin and his fiancee, who were completely frazzled and underwater with work stress and last minute wedding details, called him personally or sent a new physical invitation, I would have called him out for being a giant special snowflake.
If he had not believed my information (or version of it), and had thus called my cousin himself to check, that would have been totally fine. Or if he'd simply decided not to come even though he was invited - also fine. But if he'd indicated that he *would* go if he'd *really* been invited, but since he never received an invitation, he just wouldn't... I would have rolled my eyes so hard they fell out of my head.
OP, if you don't want to go regardless of whether you're invited that's one thing. But you actually have a reasonable amount of evidence that you likely *were* intended to be invited. If you want to find out for sure if you're invited, that's cool. And you can decline to go in any case, but declining to go because you weren't "actually invited" seems disingenuous here and honestly, may come off to others as drama. If I were the host and I *had* invited you, I would likely feel like you were just trying to Make A Point, and I would not be impressed.