Is running two monitors off a 2017 MacBook Pro 13” too much for the computer to handle?
No, it’s not too much for the 13 but it does get hot. The 15 churns out even more power in my experience, so I would stick with 13 inch models or the 2018 Air if you want less heat with the current generations of MacBook Pro. 

I would get a stand and not run in closed clamshell mode and also consider a Mac mini and an Air or iPad if you can let your desktop run cooler and not need a MacBook Pro. 



We buy Macs at work to run Adobe products as well. I have about 250 people globally that push their MBP hard all day in closed clamshell mode and we've had a handful of failures like you describe - KP / GPU halts / then need a repair. People that do not have one or someone new to running them without fail say "these are running too hot" - all 250 just run hot - especially driving pixels or doing video chat in MS teams for instance. Some even KP / GPU halt occasionally and never need repair (or haven't yet). The repair rate is less than 1% for us even when we push these hard / all day / CPU hot / closed / driving two displays.