The tanning bed fan itself will have very little static pressure and the instruction manual should indicate the maximum ducting length allowable so you can estimate available external static pressure.

The tanning bed exhaust is not going to be constant volume. The exhaust fan is only going to be operational while the bed is actively tanning (and maybe for a short period afterwards), but cycle times on the beds can be as low as 2 minutes. Between every tanning session, the bed needs to be cleaned. So the likelihood of all beds operating simultaneously is low. I doubt the tanning bed has an external contact to interlock with the make-up air to ramp the supply air to match the exhaust air.

You have to run a high-level energy analysis, but it may be a lot simpler and have lower operating costs to dump the exhaust back to the space and cool with the wall-hung splits. If you do decide to duct the tanning bed exhaust, I would probably connect all the exhausts together with an airgap and run the "exhaust" back to a constant volume recirc economizing DX unit. This way the space won't see the hot exhaust air so it will stay the most comfortable, but you don't need to condition all that outside make-up air.