If you pre-fill the new cooler (and the lines, with them pointed up in the air - 4 hands helps a lot!) and then swap all the connections around quickly, the level will be "close enough" to drive around for a while and not worry too much. However, even on the street (at least in Texas weather) that fluid looks/smells horrible after ~20K miles, so it's a good excuse to go swap all the fluid shortly afterwards anyways (either at the dealer, or DIY).
GTM has a chart at the bottom of their fluid swap PDF about fluid temps vs change intervals:
http://www.gtmotorsports.com/Manuals..._Procedure.pdf . I haven't (yet!) had live trans temps to look at, but I'd guess in hot weather even street cars are bumping into that 215 range, and track cars probably go quite a bit higher without add-on cooling.