What's the best way to get cauliflower burger patties to hold their shape on a grill?

Put them in the freezer for 20 minutes prior to grilling. It works for crab cakes.

You can put a piece of foil underneath them. I get this would detract from the grilling effect, but if you merely wanted to get them cooked while grilling, this could be a solution. I'm afraid that even with egg and breadcrumbs, they'd still be too delicate. Though I do think a cauliflower patty sounds great.

I agree with using egg as a binder. Use just the yolk. I would add panko to absorb moisture from the cauliflower. Then, the tricky part - I would mould inside an old tuna tin with top and bottom cut off (or similar size ring mould) and bbq with the ring mould.

I've seen recipes for cauliflower patties before. It seems the best way to make sure the cauliflower is devoid of all its moisture (which would probably be the culprit that prevents the patties from holding together) is to chop or process it into very, very small pieces (rice-size) and roast it on a sheet pan until it's dry and just slightly caramelized. Then, use cooked lentils and bit of flour or breadcrumbs as a binder.
I've personally never tried this, but like I said, I have seen recipes for it.

Maybe go at it a different way -- what if you sliced the cauliflower into steaks instead. http://food52.com/recipes/20792-dan-barber-s-cauliflower-steaks-with-cauliflower-puree
I'd bet you could grill them over indirect heat until they were pretty well cooked, then sear them over direct heat to finish them off. They'd still require some gentle handling but would hold together.
I'd also consider taking some notes from this cauliflower tortilla recipe to see if you could translate it into a burger: http://roostblog.com/roost/2013/5/17/spring-vegetable-tacos-with-cauliflower-tortillas
If you riced, then cooked, then squeezed the liquid out before adding your binders I'm thinking it would hold together better.

Did you mash the cauliflower before making the burger patties?

You can get baskets and screens for the grill top. It's unlikely that a home made patty would survive a regular grill top.

I think the defect is the cauliflower itself. It doesn't lend itself to any kind of "burger" patty even a faux one. While I'm tempted to suggest lard, that would probably defeat your purpose. Perhaps focus on lentils or other legumes. Eggs do make a good binder.