Air compressor makes LEDs flicker
I believe I may be experiencing the inverse of your situation 

My ceiling fan power is separate from the LED lights in the fan. If the fan isn't powered up to 100%, the lights flicker like crazy. Once it's turned up all the way... they're fine.

... so my theory is... you don't have enough juice any more. Assuming these are bulbs, if you unscrew a few, and run the compressor, do the remaining still flicker? 

I think the dedicated service for the compressor is a good train of thought. I only question sustainability over the long term and future growth. You may want to look into your current wiring set up for a possible more permanent fix.