You’re conflating how warm bloodedness works in mammals with how it would work in theropods. They were warm blooded because they could not shed heat from metabolic processes due to their volume to surface area ratio, not because their bodies needed it. We still do not understand how it actually worked. This is evidenced by us still not knowing the reason why stegosaurus had plates, Spinosaurus had their sail etc.

Also, some birds can go weeks without food during migration or injury. Further to this, theropods are many magnitudes of size larger than birds. They would have far greater fat stores than modern birds.

Disclaimer, this is based on my 16yr old zoology major, and we are in a dinosaur discovery golden age so something may have changed.