Definitely sounds like 'low dc voltage' problem.
Fan comes on but on runs for a 15 seconds then shuts down......
Fan has to get up to at least 75% of rated speed in order to close 'sail switch'. If sail switch doesn't close no further action can occur (power to circuit board, then ignitor/gas valve).
With low dc voltage fan speed not enough to make up sail switch
DC voltage is present at all times from fused position on dc dist panel. Goes from panel to timed delay relay.
DC power also goes to t-stat then back to furnace. When t-stat contacts close power is applied to fan motor. Sail switch makes up....power then goes thru high limit switch then to circuit board. Then board sends power to ignitor and gas valve for igniton of main flame.
Power is getting to motor but not to board for ignition when on battery source.
Need to check voltage at furnace when on battery source and that grounds are good, tight and clean (at furnace and from battery).
Incorrect dc polarity will also cause main flame to drop out shortly after ignition........can't hold gas valve open.