Great post, and a good idea too!
It's very simple folks... P = I x V, or spelled out, Power equals Current times Voltage. With 10x more voltage, you need 10x less current to get the same power.
The 120v pump the OP listed pulls 7.5A at 120V which is (7.5Ax120V) 900 Watts of "power" for airing up tires. That means that your 12V 2000W inverter is supplying 900W to it by pulling 900W from your truck at ~13V, assuming it's running and has a decent alternator, which works out to (900W/13V) 70 Amps of draw. If the expensive 12V pump ran off a cigarette lighter, about the most power it will have for airing up is 200 watts. (15Ax13v)
If you're willing to go to battery clamps or Anderson connectors, Dual Viair 400C's will pull about 60 Amps, which is only a bit less "performance" than the 120v setup, but dedicated only to airing up tires and such, rather than being able to also power your house in times of need.
I just ordered a 2000w pure sine inverter that I'll be installing in my truck too. I am also replicating the factory dual alternator setup to go along with it so I can get more amperage at idle or not far above. My intention is that it can do generator duty to power the house or whatever other 120v things I might need when camping or working around the "ranch".
Going with the dual alts and inverter will cost me about the same as buying a little 2200w inverter generator, but seemed more useful. I really like the redundancy of dual alts, since I had the factory alternator quit charging on me just before 200k, and the reman replacement now has over 100k miles on it...)
(Rainman Ray's Hurricane video has him using just under half a gallon of fuel per hour running his house off his idling Duramax, and we have pretty much the same year truck.)
I will keep using my single Viair 400C to air up tires, since it's already mounted to the truck, and since I don't have a small 120v compressor, nor do I want or need one at present...