There's two kinds of air-start carts used nowadays:
The less prevalent (nowadays) are actually just turbine APU's on carts basically (called "huffer carts" in my part of the world). Examples of these are units like theA/M32-60's which are basically Garret GTCP85's mounted on a cart.
https://www.youtube.com/watch?v=CUnHAv123ys - I was repairing and reconditioning these units for civil use before I got into aircraft maintenance.
They put out just as much air as most aircraft APU's and would have no problem running the packs on a 737 for example. Running both packs may push it a bit, but it's capable depending on age & cycles on the cart. Your operator, if they are using carts like these, may prohibit operation to extend the life of the unit (just like your APU in the airplane, it runs hotter with bleed extraction/load). Also - as these units aren't typically maintained to 'aviation-standard' - they may want to reduce the risk of contaminating the whole air conditioning system if the unit should fail.
The other type of air-start carts (and more prevalent now than the above units) are traditional gas or electric (or hybrid on newer units) air compressors which charge a large tank. The individual compressor is a very hefty unit, but incapable of producing a sustained 45-50PSI at the flow required to start an engine on it's own. The compressor simply charges the large tank. The pressurized tank provides the pressure & flow required for usually around 60ish seconds, enough to do a start only. Obviously, if you tried to utilize the packs with these units, you'd quickly discharge the tank. These units are more reliable, cheaper to run, and a heck of a lot quieter (for those fancy European airports with their draconian noise limitations
).
System's wise, there's nothing preventing you from using external air (different then external
conditioned air) to run the packs. It's teed-in right into the bleed system. The ECS doesn't really care where the bleed air is coming from. There is a discrete going to the APU ECU to build up the pressure when packs are selected on, or engine start is selected (as the NG APU uses a separate load compressor with variable vanes yadda yadda yadda...), but other then that - if there's duct pressure showing on the gauge, it'll run the AC packs.