My best guess is that they recommend a 200 Amp service so that when the
50 amp spa is running, there is some power left for the rest of the
house. With 100amps, it will work, but you you're taking close to half
of the capacity for one appliance. And I agree, 5hp sounds mighty high
for a spa pump.

If you can't live with 50 amps for the rest of the house, or need to
expand the panel for breaker space, etc, then it's time to upgrade the
service.