The only thing I could find is a walkthrough for this game.
There was only three mentions (I think) of the dog.

Apparently this is what happens (this is the next bit after you apparently receive the "Heart Of Stone"):

Jump down to the snowy platform below, shoot or
avoid the kid, and find the tree with the bees and the black spot underneath,
summon the beast, breathe on him and send max into the now visible cave.
Return to the grinch and head back towards the snowy peak till you drop off
the platform, turn left(this is up the left side of the wooden rail)and run
towards the left end of the house, shoot or avoid the kid on the way, as you
approach the left end of the house turn left into the alcove and meet another
guard, get rid of him, butt drop the crystalized present and send max into the
cave for another BP, return to the grinch, go to the house and leap the fence,
avoiding the dog collect another BP to the far right of the house and return
to the safer side of the fence, now watch the dog, he will follow a circular
path around the inside of the yard stopping briefly under several overhanging
snow drifts, this next task will be on a timer so think about what buttons you
need to locate to switch gadgets and complete inside of time given, shoot a
snowdrift over the dog with your egg launcher(timer starts), switch to the
slime shooter, jump the fence and run to the right side of the house, aim your
slime shooter at the three pairs of ski's and slime them, if you do this
inside the time given you will see a short clip because you just "Slimed the
Mayors Ski's". Now if you havn't already completed the "Droop all the Trees"
mission, do so now before returning to the computer room and on to Who Dump
leaving the "Replacing the Candles on the Cake with Fireworks" mission
incomplete till you have the "Octopus Gadget" and can get to the "White
Mountain Civic Center."

I hope that was what you wanted. I've never played the game so I wouldn't know if that is right or not. Of course if you've done that and you are truly near the end of the game then I can't help, sorry.

According to the walkthrough, that part is only about half way through the game by the looks of it.

Time taken to answer: 2 hours 55 minutes.