It means "It's (not the cat but) me who will be relieved" or " _I_ will be relieved". Note that she explicitly said , followed by an **exhaustive-listing ** , which signals there's a contrast between and the cat. If the sempai simply wanted to say "I'm relieved (after hearing what you told me)", this exhaustive-listing would not be necessary.
> A:
> Because you (=Sempai) have amazing boobs...
>
> B:
> Are these (=my breasts) annoying (to the cat)?
>
> A:
> No, I thought they would make the cat relieved.
>
> B: … ****
> Hmm...It's (not the cat but) _me_ who will be relieved (by holding the cat).