A .223 does not generate as much heat as a .22-250 or other larger cartridges. However, depending on the amount of shots you take and the outside temperature, a thinner barrel will heat faster than a heavier barrel. If you will be shooting 5 shots within 2 minutes at 80 degrees and let it cool for a good 10 to 15 minutes, then I don't think that the T3 lite barrel will be a problem. If you will shoot more than that in a relatively short or shorter time, then the varmint with the heavier barrel would be a better choice.
Zachary