From my experience shooting factory class 12 years and more ago, and depending on the distances you'll be shooting at I'd be willing to bet that a light barrelled .243 would likely not do too well, so maybe just seeing how it shoots the way it is, then fiddling with it (bedding, floating the barrel, etc) to see what helps and what doesn't (or even hurts) will be informative. As Boyd says just floating the barrel without bedding may well hurt accuracy.
Depends on how much time and energy you want to spend.
Maybe 20 years ago I was fiddling with a 700 Mountain Rifle in .280 Rem and had planned on floating the barrel and bedding the action. It shot very well for a light factory hunting rifle though so I left it like it came from the box outside of getting the trigger down to 3 lb. After working on loads for it the accuracy started to go south so I took it apart and was surprised to find that the recoil lug was only making partial contact with the rear of its recess, and action and barrel were trying to move to get the recoil lug snugged up to the recess. The barrel got floated and the action bedded, and accuracy returned. Sometimes it's hard to tell what a rifle will do without doing some shooting.