Why not a piston soak as well for the carboned rings? I've used
Lube Control LC-20 as an overnight remedy on a number of cars (1970's through 2001) with real success.
MolaSoak: LC-20 + Schaeffers #131 Neutra
I've also used
Auto-Rx Non-Toxic Sludge Remover and Cleaner for Engines and Transmissions in combination with above.
For an old neglected engine they do well without the problem of nasty solvents. I like them in general use as well (per directions), and also FP-60 Fuel Power (products which I and my family have used over forty years).
And, BTW, I'd guess you know that BG-44K is not the same formulation as years ago. Good, but maybe not as effective. The above aren't cheap, but far from as expensive.
My first thought on de-carboning an engine is still a coke bottle with calibrated thumb to control water flow and the other hand working the throttle linkage. Guess it shows my age. Seeing an AMPCO Top Lube bottle on a firewall was always a reminder of same.
Good luck as it goes. Quite encouraging so far.