There's nothing wrong with using bullseye for lite/plinking loads in the 44mag. Bullseye's an excellent powder but like all powders, bullseye have a pressure curve where it really shines. Stay above that minimum pressure and your golden.

Can't even begin to count the # of posts quoting 2.7gr of bullseye/148gr hbwc in a 38spl. +/- 14,500 psi
3.5gr in a 38spl pushing 158gr bullets. +/- 15,500psi
5.0gr of bullseye in a 45acp case/225gr cast rn bullet +/- 15,000psi
5.0gr of bullseye 240gr lswc/44spl +/- 15,000psi

Speer who is known for having low pressure load data for their soft swaged lead bullets lists for their 240gr swaged lead swc:
starting load 5.5gr of bullseye
max load 6.0gr of bullseye

The problem with that 5.0gr load is the bullet style/crimp isn't aiding in the buildup of the short start pressure along with it's a +/- 11,000psi load to begin with.

Take a look at your cases and see if theirs soot on 1 side of them after shooting your soft load. That's a tell tale sign that there's not enough pressure with that load to seal the cylinders. Make no mistake about bullets seated crooked do nothing but either give false reading of under pressure with lite loads. Along with magnifying the soot.

There's nothing wrong with bullseye powder, couldn't even begin to count how many #'s of it I've burned over the decades. At the end of the day your going to find that lite loads in the 44mags like:

6.0gr +/- fast burning powders like bullseye/clays/reddot
6.5gr +/- greendot/trailboss/American select/ww321
7.0gr +/- unique/herco/power pistol

Last I knew you could go up to +/- 9.8gr of bullseye with a 240gr lead swc in the 44mag. Bump your load up doing ladder tests starting at 5.8gr going up in .2gr increments. I doubt you'll get past 6.6gr to find what your looking for.