What Happens When You Exceed a Solid State Relay's Max Amps?
With the concerns you state, and especially with having the relay installed in a wall, I guess the answer is:

Don't.

You might add a fuse, but then, you'd have to find a rating in the relay's data sheet telling you if it is approved for usage with a fuse, and if so, what the fuse's rating would have to be. All in all, you would have to do the same qualification for the entire system (relay, fuse, type of outlet socket, possible loads) that the original vendor did for the combinaiton of the relay and fountain.

The relay will not limit the current (amps) to a level that is safe to itself. When overloaded, it will likely act like a short circuit for moderate overload conditions and will fail open (or explode, or burn) for severe or ongoing overload conditions.

The thing to remember is this: Once you exceed any of the ratings, the manufacturer won't guarantee anything they tell you in the data sheet any more and you are on your own.

Also, with having the relay installed in a wall where it can't really get rid of any heat it produces, I recommend using the derating curve for 60...70 °C, which tells you to rather not exceed 2...3 A.

A hacker's answer, however, is:

Yes, your setup might work, ...

... but you would have to make sure all is well and within the specified limits and characteristics.