Question: I am attempting to set up my Pi-0-w to wake up a computer on a local network. However, Wake-on-LAN is not an option. Does anyone have ideas for an alternative? Perhaps a way to wake through a USB connection?

Answer: 
If you are overheating due to high computational load, your only choices are to compute less or dissipate heat better. 

We don't have specific information to suggest how to compute less. Given what you've said, the only thing that comes to mind is to try to use a second Pi and distribute the computational load somehow. That will have marginally higher cost for the second Pi and require more engineering to handle the load distribution. 

To physically dissipate the heat, you seem to know the options: Bigger heat sink, better circulation of air, or active cooling. Active cooling, of course, is potentially expensive. On the other hand, depending on the type medicines you're dispensing, you may already have refrigeration on the machine that you could use here, i.e. if you have a refrigerator unit to preserve medicines anyway, maybe put the Pi inside the refrigerator.
The Pi3B is the most prone to overheating. If you are using a heatsink and fan there is little else to suggest without more detail of the product. 

The Pi3B+ or 3A+ runs much cooler - but Ubuntu needs updated firmware.