It is nice that you have an air compressor. I do not have space in my garage, so I bought a super duty DeWalt 20V impact wrench to eliminate the need for a real air compressor. But it is pretty hard to inflate that spare tire. In the end, I got blisters in my hand. Maybe I should buy a light duty compact air compressor. By the way, can you set up a psi value and your compressor will stop exactly at that value?
I switched out my spare for a full size on my MDX. I usually pump the full size spare up to 38-40 psi because I will usually drop a few psi every month. I can always let air out if the psi are too high when I need the spare.
I did the Viair 90p portable air compressor I use for my bikes (700X40c and 26X4 fat tires) and pumping up vehicle tires. It was about $75 on Amazon and it connect directly to my battery for power (worry about blowing a fuse with the air compressors that connect to the 12v plugs inside the MDX). The Viair guage is off; but, I use a hand held gauge and/or TPMS to double-check pressure.
Side Note: Tire manufacturers talk about tires only having around a 10 year life max even if the tread is still good. I still have my donut spares in my 06 TSX and 08 RDX (full size spares for both in the garage).