Sounds like your Onan is not self-generating ignition voltage. The little ones like yours and mine (Microlite 2800) get cranking and ignition voltage from the battery as long as the start switch is held down. Once the switch is released, the Onan makes its own. This is the same whether gas or LP; your 2500LP is the 2800 derated for LP. I'm 95% sure you do not have am oil level switch.
If you have a failure of the voltage regulator, brushes, or rotor/stator, you'll have the same symptom... starts, but only runs with the switch held down. The service manual will have a series of steps to determine what it is, but they require a volt-ohm meter (VOM).
Your most likely failure is the voltage regulator, about $225 from Flight Systems (they also have some troubleshooting steps online). On my 2800 and probably your 2500, it's a small cigarette-pack sized box, located just out of sight under the engine's cylinder. It's mounted to the engine's base pan with one Torx screw. It's out of the direct line of sight, but can be replaced without pulling the generator.
Good luck with it.
Jim, "I drink coffee for _your_ protection."
