BetaB4 wrote:

"If I do the timer idea, could the fact that the battery charger will still
be set to "ON", and will still be connected to the battery after the timer
cuts power to the battery charger, cause the battery to discharge and drain
back through the battery charger?"

The self-discharge rate of the battery will be magnitudes greater than
any discharge back through the rectifiers of the charger.

Very surprising that your charger doesn't switch to trickle charge mode
once the battery is charged. Even the cheapest chargers nowadays do
that, i.e.
"http://www.harborfreight.com/cpi/ctaf/Displayitem.taf?itemnumber=45005"
"http://www.harborfreight.com/cpi/ctaf/Displayitem.taf?itemnumber=66783"