Probably not.
So should I just email the hotel directly and ask if they have their own trainer and how much it costs per session?
Training in Hong Kong is too expensive.
Might as well go to Shenzhen, chill for a night, and find a trainer there.
It's best to confirm upfront whether the coaching fee includes the venue, so you don't end up arguing about it on-site.
I think the boundaries here are pretty blurry: hotels usually only allow guests themselves to use the facilities, and bringing in an outside coach raises safety and liability issues that hotels generally don't want to take on. Even if you book a room, the coach may not be able to enter the pool area as a visitor. The safer approach is to first check whether the hotel has a partner coach, or ask clearly whether an external coach can register as a visitor to enter.
If your priority isn't hotel facilities but finding a cheap trainer, then personal training sessions at a regular gym in Shenzhen and a hotel pool are two different things—are you planning to have the trainer come to your hotel room or a public area to train you?
The key is whether the hotel counts the pool and gym as areas open to the public. Some are open only to guests, so outside trainers aren't allowed in—even if you've booked a room, your trainer can't get in without being registered.