GFI for Fountain Pump?

Its tripping indicates an earth fault that requires attention on safety grounds.

Inconvenient, saving life, isn't it?

If the GFI device is outside and is tripping, then you need to place a cover over it.

1- Gang Weatherproof Bubble Cover

Check Leviton GFCI Accesories. Search for 5977 and 5997 series.

http://www.leviton.com/OA_HTML/ibeCCtpSctDspRte.jsp?section=10136&minisite=10026

Could a GFI breaker be used instead of the GFI outlet?

ChazL

Yes, you can, much more expensive breaker, but like the OP mentioned about the cover, if the receptacle (with out cover) and breaker are wired correctly, you will trip the breaker now. Put that cover on and be done.

Yes Jimh77,

The breaker is more expensive, but the OP did not mention anything about a cover so I was just giving another possible solution adding to the one Wareagle mentioned.

ChazL

Yes,

It is the law, contact the AHJ and ask him to approve you're solution. (you will have to wait some hours before he stops laughing)

An alternative approach is add a junction box in the feeder line to the outlet plug and connect the cable from the pump directly to the power.

You can add clocks and other futuristic devices like to organise the functionality.

Some GFI receptacles are notoriously poor quality, and tend to fail for no reason at all after a short time. It may be that you simply need to replace the receptacle itself. Adding the weatherproof cable housing mentioned earlier is a very good idea, but may not solve the issue if you have a bad GFI.

Cheers,

Steamerst

It is a real problem: the GFI becomes nervous due it's own loss and the handy neighbour takes it out.

Then the accidents happen.