Front Gate handle keeps coming off

I have a timber front gate with a Victorian brass handle...that's just screwed into the wood. About 6 months ago, a meter reader ripped the handle off the gate after my dog barked at him (from inside mind you).

Anyway, the screw holes were completely shredded...first attempt I packed them with matchsticks and rescrewed...held for a while until one of my kids ripped the handle off after the gate was a bit stuck. Filled the screw holes with bog and reattached...held for a few more months until the other day when someone pulled the thing off again!

I'm just about out of ideas...I can't use longer screws than I already have...the bog just doesn't seem to be strong enough.

Anyone have any ideas? I suppose my other option may be to actually remove a chunk of the door timber where the screws go in and refill with another bit of timber...probably getting to the limit of my handy skills if that's required.

Is it possible to mix the bog with an epoxy high strength glue?

Any help greatly appreciated.