I store my BBQ in a fiamma back box along with any water carrying kit (aqua-rol etc). Keeps all the mucky stuff separate. Never clean the BBQ when on tour, just wipe any build up of grease off with kitchen tissue then burn any residues off before cooking any food.
BTW fitted a Bulfinch bbq point earlier this year and it's had loadsa use. Rank it alongside the gaslow in usefulness and no need to carry a separate gas cylinder just to use with the bbq.
Yes I wish I had an external space too. Unfortunately any back box would overhang my parking space at home and project into the road. No room on the the roof either as it full of solar panels. I did fit a Bullfinch and agree with you over it usefulness. I have a long hose on the Cadac so I can use it either side of van for shelter. Works well. The whole kit packs away neatly to nothing.