How long does it take you to put the body on? If it's idling for any amount if time the engine will load up with fuel, and cut out (usually when you try to accelerate). Leaning the LSN should help slightly, or else blip the throttle when you're putting on the body - possibly get someone to do this for you as it's difficult to do both at the same time.

Other possibililies include the bodyshell catching on the tyres (unlikely but possible) or the shell somehow restricting the fuel flow (pinching fuel line) - this is equally unlikely due to the routing of the fuel line on the TS4 (i have one).

Running with a bodyshell will not reduce the air flow significantly enough to cause the engine to stall - although it will run slightly hotter if there are no ventilation holes cut in it.