Oh dear that sounds exhausting (((hugs)))
The first thing I noticed is that her first A time seems quite short, at 5mo I'd expect her to be going at least 2h minimum. Could you start by trying to push that out? I suspect that she's got herself into a UT/OT cycle, ie she's UT going down for that first nap (which probably makes it even harder for her to settle without the dummy, which is obviously a strong sleep cue/prop for her), and so she's not tired enough to transition into a second sleep cycle (ie takes a short nap), but then as the day goes on, all the short naps result in OT, which also causes short naps, and so it continues...
The other thing I wondered was whether her reflux meds are at the right dose for her - do you think they're working at the moment? I spotted a few very short naps where she woke after just 20mins or so, and that often indicates some discomfort
When did you last see the doc about her reflux? Is her cot inclined? Have you found that she'll sleep upright at all, eg in a sling? I really relied on the sling at this age when mine struggled to have long naps - as a reliable way to catch up on OT with a nice long nap, before going for the next nap back in the cot. I think my DS especially benefitted from that as with hindsight I think he had undiagnosed silent reflux
I just didn't have the confidence to ask for a second opinion at the time. When they're OT they have a much harder time settling, especially if they haven't learnt how to self settle before. But if you haven't used a sling up until now, I probably wouldn't start, it would only introduce a new prop...I think. Either way, if you 're going to sleep train you'll need to make sure she's as comfortable as possible, or she may just associate sleep with pain, which could explain some of her resistance.
As for PUPU/shh pat etc, they do involve some crying, since you're changing the way she's used to going to sleep. How long did you try for? Tracy recommended trying for 45mins, then coming out of the room for a 10min break or so, having really quiet time, before going back in to try for another 45mins, and if still not settled, give up and feed. Then make the next A time really short before going back and trying again, hopefully she'll then crash out! Same sort of thing for resettling, I'd try for 45mins or so before giving up and feeding. But if her reflux isn't quite under control and she's still in discomfort, don't use PUPD, as it can make the reflux symptoms worse. Instead, you'll want to work on calming her in the cot, only pick up if she's really showing you that she needs you to do that. I know it's really, really hard. Also if the crying makes the reflux worse, you'll need to do something different. But if you do want to wean the dummy, and it sounds as if that would help, you'll need to stick with it and be consistent - get support IRL too as it's hard!
On the other hand, if you think the reflux does need more management, just work on extending that first A for now, until you're sure she's comfortable. Be aware that the first time you try it she may not sleep at all, the next time she may finally fall asleep after over an hour, then it may be quicker and then she may seem to regress again, but if you stick with it, providing she's comfortable, she'll eventually learn a new way to settle. In the process you may find for example that she prefers a still hand on her, rather than a pat - I think this would be quite usual for a reflux LO actually. She may prefer a hum to a shh. She may (like my DD) need you to get your whole arm in there to snuggle up against to start with, to feel you close. But eventually you'll be able to do less and less. But anyway, the first step is to get her sleeping in her cot without her dummy (if you feel it's the right time to do this - I don 'to want to push you if not) however you need to do it. Sorry I've rambled, you may find this interesting - particularly the second post in the thread, which contains an interview with Tracy which I found really helpful at this stage:
Sleep interviews and member consults with Tracy