Hi there
Really sorry to hear you've been having such a hard time - patting for months on end in a dark room, well that is utterly draining!
I am going to suggest something quite different, take it or leave it.
I have not seen anywhere here that your LO is an independent sleeper or not - does she go to sleep by self settling for naps/BT? (it does not matter if she needs additional help for NWs, I mean her usual way to fall asleep at her normal nap and BT) Being able or unable to self settle can have a big impact on ability to stay asleep. It would help greatly to know for sure.
Having said that...
...I feel that if she has the ability to SS a longer A time may help with increasing nap length (she may still need some training to teach her that she is supposed to sleep longer), but if she is not able to SS yet then a longer A time whilst you sleep train may also lead to greater success on the sleep training front (as clearly the 3 months of patting has not shown great progress).
My suggestion would be to move directly to a longer A time of 3hrs. This A time is more suited to an older LO really but as your Lo is already doing 2hr 30 by feeling is she could be one of those who need longer...or the length will help you sleep train. I would also suggest spending a few days focused on nap 1 where you are very serious about her going back to sleep, rather than purely soothing her you are telling her it is time to sleep and staying in the room for 30-45 mins to do so.
I know you have another child to care for, that makes it harder, any chance your older one would watch a movie at that time for a few days? In the long run teaching your younger one to nap longer is going to free up more time for your older one anyway. Depends what you think and if there is someone to mind your older one for that period of time really.
With the longer A and you showing/telling her to sleep longer your routine may look something like:
Wu 6
A 3
S 9 - 10.30
A 3
S 1.30 - 3
A 3hr (or a short A time and quick CN with a later BT giving 11 hr night which looks like you LO likes)
BT 6
Consider the worst that happens, that she still doesn't sleep more than 30 mins so your routine then looks like:
WU 6am
A 3hr
S 9 - 9.30
A 3hr
S 12.30 - 1
A 3hr
S 4 - 4.30
A 2hr 30
BT 7
I've seen a few LOs adapt really well to an older routine and personally feel if mum is struggling (who wouldn't struggle with all that patting? I imagine you are at the end of your tether!) it may be a faster route to a more settle routine.
I know this is quite different to general advice, something for you to have a think about.
Many hugs and luck whatever you decide xx