You can always plug the holes temporarily (with wet clay, etc) while you dip, then remove the plugs after the glaze is dry. Or just dip away and let the glaze fill the holes, then clean it out later with a pipe-cleaner or wire.
I haven't used the triangles you mention, so I can't address those specifically. But here are some general ideas: First, if the backs are not really intended to be conspicuous and you only want to glaze them to seal them, you can put a much lighter coating of glaze on the back than the decorative glaze on the front. This will most likely mean everything is brushed on at first, but if you go into production there are other approaches possible, such as a light dip, masking the back, and a second dip for the front.
Having less glaze on the back will mean much less dripping or running onto the supports during firing. After a few trials, you should be able to get good sealing coverage with no running at all.
I suspect the kiln wash idea will be helpful with the triangles, but whatever you use you may have to accept a blemish where it touches, or else be prepared to do a little touch-up grinding. You can probably do that with a hand-held stone, no power tools needed.
But if you have glaze-free hanging holes on these masks already, the simplest approach might be to fire them while hanging on high-temperature wire, as is done with beads. You can make your own support for the wires, or string it between kiln posts or something. Keep in mind that the wire will lose stiffness when it is hot, so keep the unsupported length as short as feasible until you get a feel for this. You don't want the weight of the masks to sag the wire such that the masks touch the bottom or fall off.
For heavy-ish things (these sound heavier than typical beads) I have had good luck with bundles of wire for added stiffness. I usually coat the wire with kiln wash, which helps if some glaze sneaks around the edges and onto the wire. I find the kiln wash goes on easier if the wires are warm, so I put them in the kitchen oven first.
Hope this helps!
Bob Masta dqatechATdaqartaDOTcom D A Q A R T A Data AcQuisition And Real-Time Analysis