Just switch out the relay. I think the one for the headlights is the same thing or buy a new one for $12. Actually do this. Put the key in the ignition and switch it to "on" without cranking the engine. You should be able to hear the relay click and a hum as the in-tank pump comes on. If you're not sure, leave the ignition on and pull the relay out and plug it in again. You'll feel the click when you plug it in.
If you hear the click but no hum from the pump check the wires going back to the pump. Mine corroded off after 10 years of salty highway driving and they'd make and break depending on which way I twisted the wire. If your lucky they might have corroded at the connector so clean it up and reconnect or chop it off and solder/tape/seal the wires straight to each other. Marine Goop makes a good sealer.

Another thing: check the belt tension on your alternator, pull the connectors and clean them up. I still say your alternator/battery sounds like the culprit.