Can you strap a heat-sink to your cpu, wrap/ attach it to the blower end of an air compressor and then cool your pc?
An exhaust tube could be run out the back through the I/O port, or if you have a case with liquid cooling tube holes cut already, they might work.
I am not sure how small a form factor you can get an air compressor in, but tubing is flexible and as long as it isn't the same size as the computer it might be practical.
Any other ideas, or refinements?
TL:DR swap the fans on a cpu cooler for an air compressor blower.