I think it's less a question of size than ratio. The shorter the drum, the less resonance. The larger, the lower the note. To get a fast response AND deep (<30hz) bass sound (avoiding cardboard effect), you need a larger diameter. A hasty conclusion would be that bigger/shorter is therefore...
What the restauration will involve is not clear from your pictures. While collectors focus more on the mahogany jazz models, a complete Asba with 16 and 18" FT is a rare find. All in mint condition should fetch perhaps 1200 to 1600, but mind that this is not exactly a liquid market.
House kits pedals
Stuck butterfly nuts that did not receive a drop of lube in 10 years
Snare baskets that tilt while you're playing
Dead worn out drumheads, sometimes with hole and plaster
In other words I hate most of the London rehearsal studio kits. Still I have the...
During the 70's studios moved from live recording on 4 or 8 tracks (sometimes just 2) to complex 24 track mixes and controlling sounds with reverb, echo, noise gates, etc.. Drums were producing that 60's jazzy sound which we now call "vintage", which they kept tinkering in the name of control...
At first, I would suggest loosen the pressure of your foot on the pedal or, as was suggested above, play more on the shoulder.
If it does not work, I would suggest hi-hat with vent holes (Paiste Sound Edge or Zildjian Quick Beat)