dropping my beloved berrypaw here as a potential friend!!
berry was also born as an outsider, although she was found abandoned in a berry bush when she was very very young by a RiverClan warrior. she and carppaw could bond over that fact because part of berry's backstory is being looked down upon for not being clanborn — she wants to work hard and prove herself, though her self-doubt is enough to crush her at times.
berrypaw is a timid lil gal, insecure and introverted, but also desires close friendships and being able to trust. i could see their potential friend dynamic as carppaw being super supportive and a pep talker for berry, while berry could offer carp a lot of kindness and support for anything she's dealing with. it could be a pretty sweet and wholesome friendship that grows over time, 'cause my girl's got trust issues but desperately wants friends. she's only really close to Stagheart, the grumpy ol' tom that found her, and her former mentor Mistyshard as of right now.
bonus idea: berry's only interested in males so she can't be considered a potential mate, but if carp had a friendly little "crush" or something too, that could be interesting. berry would be completely oblivious unless carp said something tho, she's kinda aloof to that stuff and wouldn't really consider anyone liking her in that way unless they're pretty forward with it. this idea's only really if carp is looking for non-serious "crushes" rather than a single huge crush that she pursues, of course, 'n up to you