I would recommend trying chlorine in the hot tub first. The reason I suggest this is because if you start the spa as a chlorine spa, you can turn it to a bromine spa by dumping in bromine at any time. You can't go the other way though without a complete water drain and refill.
A chlorine spa doesn't use more chlorine in ppm than a pool does. What might be an issue is CCs, or combined chlorine. These are produced while breaking down waste, and a hot tub has more waste due to high temps and sweating concentrated in a low volume compared to a pool. CCs are generally irritating, I imagine moreso to someone with asthma but this is just an uneducated guess.
CCs are what you smell when you smell chlorine in a pool, especially indoor pools. These are often actually a sign of too little chlorine, as chlorine breaks down CC's. All that said, at it's worst, my chlorine spas have never smelled as bad as an indoor pool area often does.
And again, if it does bother your asthma, then switch to a bromine spa at that point.
Chlorine and bromine both work fine at high temps. The sun isn't a concern for either type of sanitizer for standalone spas, since they are covered when not in use. Occasional exposure of the spa to UV light in sunlight is actually good and helps break down CCs.
In summary, I'd give a chlorine spa a go first, and only switch to bromine if you have an issue with the chlorine. If you have any questions with setting up and maintaining a chlorine spa, let us know. We recommend 30 ppm of CYA is added to standalone chlorine spas, and maintain the FC level based on the FC/CYA chart: Chlorine / CYA Chart - Trouble Free Pool (so target 4-6 ppm FC with 30 ppm CYA)