You should have the engine running anyways, and preferably connect your compressor directly to the battery.
Modern vehicles are loaded with electronics and they do not "like" low voltage at startup.
This is one reason that batteries used to give "warning" before they failed through slow cranking.
Now, when the key is turned, the computer simply won't boot up so you get no spark and no command to activate the starter.
Compressors draw a LOT of current. Not as bad as an electric heater, but they do draw a lot. Many, even though they have a lighter plug, can draw enough to blow the fuse.