I am not going to say... do not use bore butter. Too many people use or have used it with good results. But there are just as many or perhaps more that used it and decided that they did not like the effect of the Thompson Center Bore Butter Wax product.
If you do not apply bore butter correctly, next year you will have a rusted barrel. If you do not clean the bore butter out of the bore before each use, and let it build up as in "seasoning a bore," your accuracy will suffer. Why would you fill the rifling grooves of the barrel with wax, and then expect rifle VS Smooth bore accuracy.
T/C #13 is nothing more then natural water with a natural soap, and it will not protect your barrel from rust. It will clean fowling out of the barrel, but then so will your dish water if you want to save that and put it on a patch.
I like to water bath the barrels. perhaps a solvent patch or two, then an alcohol/car windshield washer patch, then some dry patches and end the barrels protection with a quality gun oil like Birchwood Casey Sheath.