Does a larger Royal Navy frigate need a higher-ranking commander?

Would the commanding officer's rank differ between a normal frigate and a larger frigate?

No. Both would normally be commanded by full captains; the difference is usually the captain's prior command experience.

Both an ordinary Royal Navy frigate and a larger frigate would normally have a full captain in command. The service does not need a different nominal rank merely because one post ship is larger than another. Experience supplies the distinction: an officer may command a small ship first, then a medium ship, before receiving a major command. The same rank can therefore cover different levels of responsibility while the appointment history signals the captain's preparation.