I don't see my battery as being useful power for this unit. YLB - RailSounds Battery Replacement was specifically designed to replace the RailSounds battery in conventional, TMCC, and Legacy locomotives. It's operating profile is customized to those uses. It charges whenever power is applied and is ready to deliver up to 200ma of power for 12-15 seconds for the RailSounds board.
What you're looking for here would seem to be a simple track powered power supply, one could just press one of the fairly common AC-DC power modules available on eBay for this use, these are typically in the $5 range, and accept track AC and output your choice of DC voltage.