Go with the 240 gr bullets and 2400 powder.
The Older I Get...The Better I was...
Team 444 Member #175
Team 35 Member #2
Team 39 Member #10
Marlin League #26
338 MX...It's not your father's lever action!
A fun load for me--3 caliber size round lead balls in the .444 case, lauched by 26 grains H4198. 3 hits/shot on target out to 25 yds. Another fun load is Beartooth bullets 405 grain Flat Nose with 26 grains of H4198 ( hardly a plinker to me as it is whompum load without so much felt recoil). Paco Kelly at leverguns.com says any .44Mag load will be a lite load for the .444. The light weight Barnes bullets of 200 grains would be fun but if plinker comes from low -cost then Barnes don't fit that label. But, a 200 or 225 grain cast bullet at about 13-1400fps would certainly fit that label.
Team 444 member #205
I like the Trail Boss loads also. Great for starting or spending a full day at the range. Most folks cant shoot 200 rounds of full power 444 but with TB powder they can safely take you from 44 special to 44 mag speeds and when you want to step up just drop a full speed load. You might even get lucky with your hunting load zeroed at 200yds and your plinking load very close at 50 yds.