Get a vet to do it. Start with a lower dose than normal, if it isn't enough a second dose can always be given.
My 25+ TWH has always been drug sensitive (half tablet of Bute drops him to the ground).
He has Equine Metabolic Syndrome, which further upped (or downed) his drug sensitivity
When the vet did physicals, a few weeks back, this horse needed his back molars worked on. The vet did him first because he needed tranq'd. That way, he could do the physicals on the other three while watching this guy, who needed so little Happy Juice that it wouldn't have phased the whiskers on a Mini
My horse couldn't even hold his head up, so the vet took the halter off him, and had the vet assistant rest Duke's head on his shoulder while he filed. No speculum either and that means no twitch.
As much as you fuss with Lizzie and as calm/sensible as she is, she should not need to be twitched.
Just have the vet drug her, for sure, and ask him to do give her a little at a time. If he could stay an extra 30 minutes to make sure she comes out of the Happy Juice ok, that would be even better