Can I re-install the previous version of the software on my iPhone SE to overcome a battery issue?

Hi, has anyone here ever succeeded in getting help from Apple to put their iPhone software back to the version it was a few months before? I have a 9-year-old iPhone SE which has worked fine for many years until a month or so again when the battery appeared to go into a terminal decline. In response to this, I could just buy a new battery but I am not convinced that is the problem - I contacted Apple Support last week and they ran all sorts of tests and assured me that my battery health was within acceptable limits, however it obviously isn’t working. After a trip away over the weekend, I put it on charge last night and it was showing 100% early this morning so I unplugged it and I went back to sleep but when I woke it was completely flat. It’s not consistent though, sometimes it will hold its charge for hours at a time - it was at its worse when we were away over the weekend in an area with poor reception. My wife found in the same location that her modern iPhone ran down much more quickly than usual but when we have visited that same low-coverage location in the past, we’ve never noticed any particular impact on battery life. This leads me to think that something has changed in the latest release of software such that it uses far more of my elderly iPhone’s power to do fairly basic things. Before I spend money on a new battery, I would like to go through a process of elimination and get my phone’s software back to the version it was running around Christmas-time, before all this started to happen to see if it works properly under those circumstances. Does anyone know if that is even possible? Will Apple help me install an old version of the software do you think? Thanks.