Based on GRJ's comment I did a little research. A Li-Ion battery solution is quite feasible. In fact, building your own "CupCam" from scratch is quite easy using off the shelf components, perhaps for <$30.
Several details can be gleamed from the Lionel parts diagram (below). The configuration is fairly straight forward. There's the self contained camera /WIFI unit, an on/off switch, a regulator module and an LED board.
The "OEM" camera runs on 3.7 V. Perfect for a single cell Li-Ion battery - the popular 68650 battery size. One could replace the existing Lionel voltage regulator with an adjustable AC/DC regulator, a 68650 Li-Ion battery and a 1S Li-Ion charge unit. EDIT:
Given the Camera comes with a battery a separate 68650 cell is not needed.
The camera would hook up to the switch as currently designed and the Li-ion battery. A single-cell 68650 battery holder would finish the installation. Indications from the Camera offering indicate a 68650 battery with a 2700 maH rating would run the camera for 4 hours w/o track power.
It's possible the existing Lionel regulator module can be reused in place of the additional regulator module. It depends on the Lionel module's output voltage (e.g., ~5+ V).
Safe Harbor statement: Improperly designed Li-Ion systems are dangerous (remember the Hover Board fires?). You need to insure your battery and battery management module are compatible with respect to over/under voltages & currents.