
How can they charge for boarding a dog that isn't with them?

Hopefully your dog will be found - you must be out of your mind with worry. :(
In your position I'd be so angry I'd see a solicitor, and start any compensation claim (even if my dog turned up) with the cost of a lab puppy (£500 ish?) plus how much he's cost you since you've had him - food, vet's bills etc. Then I'd add a sum for emotional distress - the sky's the limit there. You could never get adequate compensation but they sure as hell need a rocket shoved up where the sun don't shine, to make them take their responsibilities seriously.