Hi there, sorry you haven't had a reply sooner. I think you are spot on with the teething. Teething can cause real problems for some LOs and their sleep, but not always cause any issues in the day when they are busy and occupied.
The fact that pain relief works some times and that your DD won't let you near her teeth to put on gel would certainly suggest that there is some pain going on there. Teeth can take ages to come through, it seems to take weeks for my DD and then just as one comes through another one starts!

Try to keep medicating before BT when you think she is suffering and use the gel (if she will let you).
You are quite right that feeding her when she wakes could soon become a prop, so she will need the comfort to get back to sleep. If you are OK with that and prepared to wean at a later stage then by all means use that, if not then I would try to avoid feeding her in the night and try to settle her with a hand on her back or your voice. Obviously if she is very upset and you can't console her then you will want to PU and give her a cuddle. Just try to PD before she is asleep again to avoid her becoming dependant on you.
Teething can be a struggle at night and sometimes you just have to go with it and get by as best you can, trying to keep the APOP to a minimum. HTHs.