@GetOffMyLawn, how did you get on?
Just posting this in case it's helpful to anyone else reading (bear in mind that I am in New Zealand, so drug mentions, advice I was given and cost may differ depending on where you are).
The pill I have been prescribed is called "Cerazette." It is a progestagen-only-pill which apparently is the only option for my age (late 40s). It is not 100% guaranteed to stop periods, but has a very high success rate of doing so. I have been instructed to take it on the second day of my next period and continue on it without stopping until for about three years, and try stopping at that point to see if my periods have completely gone (if not, continue taking it). Apparently for some, there will be light spotting and a bit of a 'blah' feeling, but nothing compared to what they might have experienced without it. So, it is essentially masking the signs of perimenopause.
HRT here is very expensive, and apparently the only type they prescribe is for hot flushes (the least of my worries), so it wasn't really an option anyway. Cerazette is partly-funded ... I will need to pay NZ$50 every three months from my own pocket (which I'm sure will be worth it).