A few thoughts.
* First off, is it a surprise to anyone that Claire's a bit high strung? And even mellow folks can have bad days. Claire's a long, long way from mellow.
* Secondly, would it be that much of a surprise to anyone if Claire's insecure about this situation? This relationship with Marten is a NEW one, and may well be her FIRST one. He's living with another woman, someone for whom Claire knows he's had feelings and has been told has had feelings for him, someone whose interactions with Marten everyone in the social circle knows ALREADY split up a relationship of his. That woman's just broken up with her own SO, she's just lost her job, she's got a substance abuse problem, and that's the sort of situation where ladies in Faye's position often go trolling for some man, ANY man, to help them feel better. Hey, look, there's one right there ...
* Then factor in that Claire's trans. It'd take a woman of iron self-confidence, in her shoes, not to wonder for even an instant whether Marten would rather be with a "real" woman and is only with her because he's settling for what was handy. Hey, look, there's one right there ...
* As far as Marten not wanting anyone else around in dealing with this ... I've been in that position a lot of times. I've got a turn of phrase that's filtered into our social circle: "Non-players off the green." It's code for "I've got this, I don't need any help, I don't need anyone else mucking this up, this isn't a spectator sport, and I don't have the time or the mental energy to spare from caring for my friend to explain what's going on to kibitzers, even if I was inclined to play fast and loose with my friend's confidences and troubles, which I am most certainly effing not." There are those who don't find this endearing.
* Finally, look, folks. We've got a third-person omniscient viewpoint. WE know Faye isn't trying to get into Marten's pants. WE know that they're not banging. WE know that -- at least from what we've seen over the dozen years of the comic -- Marten's not a cheater. WE know that Faye has a serious alcohol problem. WE know that she's really messed up over Angus leaving. WE know that the overwhelming number of Marten's friends being women is because he likes having women friends, and not because he's after a harem. WE know that Faye's hospitalization was no petty matter, and that she was in serious danger.
Claire doesn't know these things. Not for a certain fact. She has no idea that Faye is as bad as she is (and it is, after all, a hard-drinking social circle, where a number of them routinely get blitzed). She has no idea that Faye isn't trying to get her hooks into Marten, and has no idea that she doesn't already. She doesn't have a camera trained in their apartment, so that she's got proof that the crises Marten claims are happening are actually happening. She wasn't privy to Faye's medical records.
If any of you think that Claire's got the serene self-confidence in both herself and her new relationship with this guy she doesn't know all THAT well to be as sure of the reality of the situation as we are, you're not paying attention.