My inflatable hot tub says "don't use with an extension cord"... but can I?
I think you have likely answered your own question. The use of an extension cord could cause a voltage drop, especially since the factory cord is long already and the spa probably has an electric motor that would be damaged by voltage drop. Also, cord/plug connections are not weatherproof.

This is a CYA (cover-your-arse) by the manufacturer. Your plan of using a short heavy duty cord (same wire gauge as the factory cord) with the connections indoor should be fine.