I don't know which spawners are in Lucky Palms. You can find out which fish are spawned by which spawner
here, although I don't know how useful that will be.
The IP fish need to be caught by diving. The ITF fish require visiting the appropriate future. Some of the other fish can be obtained by fishing in open water, as I said above. You'll have to Inspect Water to find out exactly which ones, all I remember is that some (all?) of the LN fish can be obtained that way.