I think there are a few questions about this setup that still need to be answered. One of which is simply what kind of security setup is he using. Did he/she implement a perimeter based setup or one primarily based on motion.
When you think about alarms normally there is only a need for three states and they have nothing to do with location it is about presence. Your armed/home or armed/stay is used to disable motion sensors and lean on a perimeter based setup for protection. Armed/a way is simply turn on everything for detection for intrusion including motion sensors. Then of course disarmed.
If you think about it simply as presence there is nothing saying armed stay shouldn’t be used regardless of weather the op is in the gym or main home structure. The only problem with it is if there is a concern the home protection is inadequate without the other sensors. If that is the case though I would suggest upgrading the sensors to improve perimeter protection.
Adding a second hub for the gym could be a very reasonable solution as well as. Having it’s own SHM setup could allow for the added flexibility the op wants though it would mean a separate setup to maintain. I would guess that there is a decent chance it would work pretty well without to much hacking to maintain. Some devices that are cloud based can even be seen by two hubs. I had a WEMO plug for instance attached to a ADT Security panel hub and a Nvidia Shield Link hub. It seemed to work fine. That won’t work for Zwave or Zigbee devices, but could work for cloud based stuff
With all that we are talking about smart things and I am sure a way can be figured out to send notifications, it is likely going to be either custom rules in SHM, or notifications completely outside of it.
The easiest solution will likely be to use two hubs each with their own SHM, but I would still argue if you have good perimeter based security simply use armed stay though.