Plexiglas has a tendency to become brittle over time and via exposure to a number of elements, and crazing like this is not uncommon. Your options would be to:
A) Do nothing, and this may very well be viable since its leaking into the overflow and should have no effect on the wash itself
B) attempt a spot repair similar to what you've described, though you would have to make sure you have adequate access to everything in order to get just the right amount of adhesive where it needs to be, and none where it doesn't.
C) Consider replacing the entire assembly, with guidance from a company like Tap Plastics (don't know if they exist in your area)
The biggest things to consider at this point are wether this is a repair that needs to be made, and wether the materials you commonly have access to are appropriate for such a repair. Many plastics do just fine with some adhesives, and not so well with others (IE, some adhesives will chemically melt some plastics). I wouldn't move ahead on anything until you've done some more digging and found out exactly what material you want to use for your repair, if you deem it necessary.