But I'll connect the tower only to the UPS
I see no reason not to connect at least your primary monitor (so you can see what you are doing if you lose power) and your network gear to the battery side of the UPS too.
that's 80W on top of my 860W PSU.
While it is important to take into consideration the maximum demand a PSU could pose, it is really the total demand of the components connected to the PSU that matter here.
That is, if the computer (motherboard, CPU, GPU, RAM, drives, USB devices, fans) all demand 300W, the PSU is going to deliver 300W regardless if the PSU is a 400W PSU or a 1000W PSU. And that PSU is going to pull from the wall just 375W (375 x .8 = 300), again regardless if 400W or 1000W PSU. And that would be even less from the wall if the PSU was more than 80% efficient (which I bet yours is).
Yes, inrush current matters but that is only when the PSU initially powers on. And it is what happens at the exact same point in time that really matters. Your monitors and anything else connected to the UPS are not going to be demanding maximum power at the exact same point in time as the computer PSU. The inrush current demand, for example, will be long past before the monitors start sucking down their maximum demands for power.
If this were me, I would connect my computer, both monitors and all my network gear to the battery side of the UPS. Then I would fire up my browser and open a bunch of pages. Then I would yank (well, gently pull) the UPS power cord from the wall and see what happens. The UPS should kick over to battery, and start yelling at you. Plug it back in and it should transfer back to line-in power.
Then power off your computer as you normally would - or if like me, let your computer and monitors go to sleep. Boot/wake your computer as you normally would. I bet it boots/wakes just fine.
Power off or let it go to sleep as you normally would, then yank the UPS power cord from the wall again. Plug it back in and start booting your computer. I suspect it will boot normally.
If you were talking about a much smaller UPS, I would have the same concerns as you are having. But with a decent 1500VA UPS, it should not be a problem.