I have berry cuttings and poison oak cuttings, ivy, that I don't trust to die alone, and I cram them into a silver garbage can with a lid and put it in the sun for a couple weeks, and it bakes them to death.
It seems like the Weber would be a bit shallow to get a real fire going in there. I have cut down my berry cuttings into kindling-sized pieces and they start a fire like you wouldn't believe, once dry. I can't imagine they would burn any hotter than lump charcoal, but you don't want lump charcoal ashes going into your garden soil. When I was a kid we had a burn barrel in back, an old oil barrel with a few small holes punched in the bottom, set up on bricks, with a screen lid, and things could burn inside there without the flames getting outside the barrel and threatening anything around it, or blowing flaming ashes over to other flammable things. You'd need some kind of screen over the Weber, and if you kept adding pieces, you'd need to be sure the burning ashes didn't get away and smoulder something nearby.
"Life flows on within you and without you"...George Harrison
Coastal California, USA, Mediterranean climate - no summer rain, a little frost mid-winter