Unless Claire were in one of her "moods" to begin with. If she feels thwarted, she can be...self-centered.
OTOH, I just don't see that lasting, and Claire would be mortified, later. It's very difficult to project what kind of relationship insecurities Claire may have, as she's never had a relationship before. But based on her previous actions, she just doesn't run hot for long. Jumping to negative conclusions doesn't seem her style, though I may misremember.
That said, Dora had reason--good or not--to view Faye as a rival. (Comics 4 thought 564, to be exact.) I just don't see Claire having the same issues.