How many rounds a year do you plan to shoot? Where did you get the idea you'll burn your barrel out? Your shooting at 14" gongs at 300 yards that is nearly a 5 MOA target. I'd imagine you would get a solid 10K rounds down your tube before it went so far south you couldn't hold a 5 moa group at 300 yards.
Your looking at an average of $20-30 for twenty rounds of .243 depending on what your rifle likes. I don't imagine you will shoot more than that a week on average throughout the year since that is $1,000-$1500 per year if you buy factory ammunition. I'd imagine you'll get many years of shooting at least a minimum of five, before you burn your barrel out. There are many well used .243's that that have been shooting for many of decades without burning out the barrel.
I have burned out a .243 already. 3500 rounds shot at a pace that the barrel was kept warm but not hot, and it was shooting worse than 5 moa on a rest. I shoot a lot, easily more than 100 rounds a week of rifle alone. Usually I buy ammo in bulk and bring people with to help the cost. What ever rifle I get will be shooting at least 2000 rounds a year, so it needs to last.
Oh the .17 hmr, I love the caliber and how you can buy a marlin/savage and shoot bullets through the same hole at 75 yards. But I ruled it out at the beginning, as in order to do that you need to clean the barrel every 5 shots. Very dirty round and a very small barrel.