The Savage model 16 or 116 is a standard or magnum contour barrel, depending on which caliber you buy it in. It is not a heavy barrel.
I would probably start with a rifle such as the model 12 varminter low profile, you could always upgrade parts later if the urge hits you lol, but this rifle is one of the nice ones (therefore read expensive). The "cheap" Savages are not offered in heavy barreled .243s. Probably around $800 on average.
I might have to recommend a 180 degree change in direction here, and say check out the Remington SPS Varmint. It may leave a little to be desired, but it is offered in .243 and the price will run you about $550 on average.
The Savage could be gotten cheap, but the heavy-barreled part (which I like also), kicks you out of the ballpark with Savage. All the varmint rifles are offered in varmint calibers only, and not in "universal" calibers.
Good luck, sorry we led you on. Let us know what you decide.