Rain garden as grey water dump

Could I use a rain garden for a grey water filter? For a rv producing 10 gallons a day

You'd have to check what your local rules, they vary widely from one place to another.

But my 'guess' is probably not.  If you are allowed to use grey water, most places require that you apply it below the surface, usually in a mulch pit, etc.  They almost never allow it to be applied to the surface and especially not in a puddle or pond, not without being treated first.