I'm pretty sure I've seen carry-on sized pet carriers with live pets in first class before, not 100% but I seem to recall a couple times.

I did see a lady moved from 31B bulkhead on a MD80 to 13A because you can't be in a bulkhead with a pet. Watch out for the first row of 1st then (often the least desirable seats). If you clear and only get the first row, they will be forced to move you. If that's the case, you better hope someone else in F is real nice. I'd really struggle to switch from a non-bulk to a bulkhead seat and I'm generally really considerate about swapping seats. I am however, a dog lover so I bet I'd do it without much hesitation.

Good luck.