Do you have any idea what year your 1936 cab is? 80's or so 1936 cabs are supposed to be plywood, with MDF backs. Sometime in the 90's they changed to all MDF cabs. If you have an all MDF cab, I would just sell the cab and move on.
I built a copy of the Marshall 1936 cab many years ago and I also have a Marshall Silver Jubilee 2556AV. The Marshall 2556AV is plywood, with an MDF back. I recently replaced the back with a 5/8" birch plywood back and noticed an improvement in sound, it sounds more punchy and tighter now. I wanted to change the jack and didn't want to modify the original back, so I made a new one.
Caulking the cab is a good idea, so is adding bracing. But I think changing the back, putting in thicker gauge wire, and changing the speakers will give you better results.
In my 1936 copy, I have had CL80's, WGS ET-65's, and currently have Celestion anni G12H30's. With the G12H30's, the cab was too boomy, so I made the cab opened back. I highly recommend the CL80's with this cab.
With my 2556AV, I currently have WGS ET-65's, great speakers in that cab.