I have some serious misgiving about trying to load an overbore cartridge (including the 243 Win) to subsonic velocities. There are many opportunities for mishaps here.
Generally, subsonic loads like bullets that are shorter (lighter) than what works at normal velocity levels.
My suggestion would be to go with reduced capacity cases, to prevent all the possible dangers that can damage your rifle. Unfortunately, these are expensive.
A 303 or similar rifle converted to a handgun case (e.g. 32 S&W Long, 32 H&R Mag, 327 Federal?) and loaded with +/- 160gn cast bullets should give you many hours of quiet plinking and varminting pleasure at a very low cost per shot.
I suggest 303 as you can use the original barrel and just set it back and re-chamber. This should keep the cost manageable. I imagine you are too much in a hurry to go through that process now?
My own low-cost low-noise rifle is a 44 Mag built on a Lee no4 action salvaged from a scrapped 303. The 44 case head fits the 303 bolt face perfectly. Probably a bit too much punch for your purpose.