Any reason to put a GFI outlet on the exterior for the pool pump to plug directly in to?
There is no electrical reason to do this, as long as your existing exterior outlet is on the protected side of the internal GFCI.
Is double stacking GFI outlets on the same circuit an issue other than troubleshooting if something trips?
You might see false positive trips if you double stack, especially if the downstream GFCI is a fancy, new model that periodically does a self test. Other than this false positive nuisance, there should not be a downer when it comes to safety. You will be protected against ground faults at the pool with one internal, or 2 stacked GFCIs
My advice is to go one of two routes
1- Replace the internal GFCI with a high quality GFCI outlet if the existing one is more than 5-10 years old. Just let the outside outlet sit downstream of the internal GFCI.
2- If you really want a GFCI outlet outside that you plug the pool directly into, then run a separate 20 A circuit to a weather protected box with a new 20 A GFCI in it. You won't be more protected, but you will be highly immune to false GFI trips, and you will know where to look if the pump stops running.
PS: this should not be considered professional advice
H