What your proposing is exactly how it is done. Here's mine
  • House Battery - starting circuit and all factory accessories - as sent out from factory
  • Stereo Battery - Powers 2 large Kicker ZX amps, Xantrex 1000 watt inverter.
That said i run the Blue Sea System 7650 ACR setup to automatically charge batteries from the alternator. I personally have yet to see the need for an onboard charger. I leave them in all season and remove off-season to trickle charge. BUT that's just my setup. Others have the same, others have onboard chargers.

You can hook the leds up to either as the current draw in LEDs is measured in milliamps/Ma which is so small it almost won't register.

If you're going to add an inverter I would highly suggest you add a thermal breaker of appropriate amps remote mounted inline before the battery. That way you can enable it when needed but disconnect when not, no chance of back flow or draw.