Hey there, this age can be so tough with sleep, teething etc
Usually it's the canines they're working on, but mine's about 6mo behind on teething, so has only just got her first set of molars - are these the ones your DD's getting now, or does she already have those? If it's canines, they're quite notorious, I'm afraid. Hard for me to say from a distance how much help she needs from you, but you can certainly give her ibuprofen (and paracetamol), if she's really bothered by teething pain around sleep times then you can give it to her half an hour or so before, so that it's working when she needs it. Sounds like teething is probably at least partially to blame, especially since she's asking for the gel in a specific place. But beware of putting everything down to teething - I remember doing that with DS, thinking it was all teeth when in fact he just needed a significant routine tweak and had actually decreased his overall sleep needs.
You could try putting her down for BT even later in the hope that she'll eventually fall asleep earlier, but if that doesn't really work then it may be that you'll need to cap the nap further, in order to get more night sleep and for her to feel more refreshed. Many of us find that they'll end up sleeping even longer at night than we've shaved off the nap - eg cutting it by 15mins could result in 30mins more night sleep - or even longer. So then they end up getting more sleep overall than before it was capped. Not saying this is the magic answer for them all, just that we spent many months with DS doing just what you're describing, before we were brave enough to cap
Btw, totally in the same place with DD right now too
just not brave enough to push BT enough as I know there have been other issues disrupting sleep for her, but deep down I think I also do just have to be brave...
Oh never properly answered the wiwo thing - I'd probably try medicating first and tweaking the routine a bit and see what happens, if you feel she's happier after that and not really needing you then just go for it - yes she will be upset but if you want to wean the patting then you'll need consistency, and as I think you said, she may just get more upset with you there and not patting. Of course, she's old enough now to understand so much more, you can explain to her simply that you're leaving but will come back if she needs you, that you're just in the living room (or wherever) and need to xyz and that she can go to sleep without you, that she's safe etc - whatever explanation you feel would be most appropriate for her.