@TheoV I think the orange bill confirms it is a female northern cardinal. Pyrrhuloxia have yellow bills (though otherwise they are very similar and I have a hard time telling them apart myself).
@Arizona Docent Thanks a lot for the correction ! I wasn't aware of the bills' difference between the two species, it comes in handy. The aviary also contains northern cardinal-pyrrhuloxia hybrids so I was even less sure of my identification