[ooopdddddd]

It's not quite that easy. For the simplest example, look at https://enclose.horse/play/dlctud, where the naive solution will waste two walls to fence in the large area. Obviously, you can construct puzzles that have lots of these "bait" areas.

Like the other comment suggested, running a loop where you keep adding constraints that eliminate invalid solutions will probably work for any puzzle that a human would want to solve.

[Macuyiko]

Oh I see what you mean now, indeed:

    Score: 7
    ~~~~~~
    ~····~
    ~·~~·~
    .#..#.
    ......
    ..#...
    .#H#..
    ..#...

However, I think that you do not need 'time' based variables in the form of

    reachable(x,y,t) = reachable(nx,ny,t-1)

Enforcing connectivity through single-commodity flows is IMO better to enforce flood fill (also introduces additional variables but is typically easier to solve with CP heuristics):

    Score: 2
    ~~~~~~
    ~....~
    ~.~~.~
    ......
    ......
    ..##..
    .#H·#.
    ..##..

Cool puzzle!