I'd say building cage traps that capture the trap-avoid. You can set some up at the entrance to your fort to snag those pesky thieves/ambushers. You can also capture trap-avoid forgotten beasts. I've never found one, but supposedly you could capture a web-slinging forgotten beast. Not sure if you could turn that beast into a silk farm, however... I imagine it would be ludicrously valuable to make FB cloth. Even if it has a stupid value modifier, it would be supremely satisfying and dwarf to "tame" the great and mighty forgotten beast and use it to make socks on demand
To web the traps, you build a trap on top of the web. So construct a mini-silk farm and have a goblin (or a cat if GCS is hostile still) pass by. Then close the bridge/remove hostile creature and try building traps. conversely you can build walls around the webs to force things to path past them. You could also build some bridges to raise/lower. to close off your normal entrance/web trap entrance so giant elk birds don't trigger your webbed traps. Thus when that trapavoid FB finally does show up, you can flip a switch and have it pass through your webbed trap corridor.
This isn't hte only way to trap a trapavoid, btw. Stunned/unconscious creatures loose their trap-avoid as well. The best way to do this is set up some traps around a pillar supporting a floor above it. This will generally only work on a building destroyer though (goes to smash support, gets cage trapped when floor collapses on it while standing on trap). You can have a long trap lined corridor with a single support holding up a roof above it, then pop a lever to cause the cave in for something that isn't a BD.