I did exactly what nostr:nprofile1qqsfye65t5530wq0wpllmdz23luhjxp9drhhh2syae6kk8cp6n3k3zsppemhxue69uhkummn9ekx7mp0qyghwumn8ghj7mn0wd68ytnvv9hxgtcdspxda said and flew quite a few times without an issue. However, one flight on Allegiant did refuse to let us on and that makes for a shitty day.
American turned out to be the best because they give your dog a "known traveller" number you can enter when you biy the ticket.