I have been following your quest for improving your casting to include efficient use of your lower hand. I have followed a similar path and have experience that it doesn't come easily. I have had students that say "I can't not do that" when faced with controlling the forward thrust of the top hand.
The top priority in fly casting, both single and two hand, is to create a straight line rod tip path to maintain a straight line without slack in the cast. In both single and two hand casting this means maintaining a pivot point that is located at the top portion of the hand between the thumb, index and long finger. In the single hand cast the power for rotation of the rod at this pivot point comes from the squeeze of the ring and small finger against the palm. Many single hand casters don't use this pivot point but instead use the wrist or elbow which causes an arced rod tip path and wide open loops.
In two hand casting the pivot point is also at this position in the top hand but the power comes from the bottom hand instead of the ring and small fingers. Those single hand casters who use the wrist or elbow pivot point naturally bring this muscle memory to two hand casting. Other than this difference the single and two hand casts are the same; same firing position, same in making the rotation of the rod at the end of the cast after moving the whole rod forward to straighten the line before making the rotation and stop. To change this pivot point by pushing the upper hand forward creates an arc of the rod tip instead of a straight line. The top hand with the pivot point can be moved forward but the same ratio between the top and bottom hand use must be maintained for the straight line path.
I have found in my practicing two hand casting that my most valuable drill for developing the upper and lower hand ratio (upper hand pivot, lower hand power) is two hand overhead casting. The firing position and forward cast is the same in all of the various spey casts regardless of singles, snaps, doubles, pokes, etc. The following video is a practice cast where I am false casting 80' of line/leader. Note that I use my top hand for powering the back cast and the loop is irregular compared to the forward cast where the power comes from my bottom hand and the loop is straight and narrow. The video is helpful for correcting this error.
So I suggest that you warm up your casts with some overhead casting where it is easier to focus only on the firing position and cooperation between your top and bottom hand. This has helped me in my unending quest.
Hope this helps.....