As I recall, my sump pumps are 1.5hp, and will lift water 13+ Ft and then throw it.
The question is not how far you need the water to go, but how far upward it needs to go. Im guessing that you don't have more than 3 or 4ft of vertical lift needed, so 1hp should be more than plenty.
...yes, there is a lot more than this that goes into the equation, but if you just need water to move, I expect it'll be fine.

I'm more concerned about the heat source. Basic wood fires exceed 400F (maybe 600-700F at the hottest part, hot enough to soften Iron but not hot enough to melt it away...those classes were a long time ago...)
- Do not let the water get stationary in those pipes.
...better yet, adjust heights to ensure 'natural circulation'

taking this thought another direction: Assuming you start the pump before the fire (I guess that should be obvious) you may consider a valve somewhere inline that lets you throttle water flow through the fire - this allow you to get the water hotter before it goes into the hot tub. Dicey game to play, but good installs will have isolation valves for pumps and heat exchangers anyway.

-Not a mechanical engineer....anymore

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