The new system appears to work, though there were problems replacing the lever with the button (it's a button-press to flush - weird). These problems were twofold. First, the old one was glued in place (thanks, previous owners). Second, the button came with three successively larger shims, to adapt to any hole. Even with none of them deployed, the mechanism was still too large to fit through our hole[*], so I filed the bastard down until it grudgingly went through.
[* Insert your own joke here. Go on; you know you want to. ]
So now the bucket is back in the cupboard, and the toilet is flushable. Hurrah.
Only problem is that, through the afternoon, a headache has been building all through the process. So, probably con-crud. Bah.