[villagetinker]

Interesting, we have a south facing lanai, with hot tub and cat, and NO mold. One difference, our floor has the cool step(?) coating and then a matching paint(?) same color as the house. When we moved the hot tub after about 16-18 months, there was minor staining under and around the tub. This was due to water dripping off the top when it was opened, and from bathers when they got out, and was expected.
The mold would suggest that the area is wet/damp for an extended period. Can you check for a small leak from the hot tub? You may need to drain, and move or raise the tub for a suitable inspection. You may be able to remove all of the side panels and inspect the wooden bottom of the hot tub for a leak.
I am sure if you do a Google search for pet friendly mold removal, you will find suitable products, but I am not sure this is going to solve your original problem.

Hope this helps.