I am on Zoely and I love it.
I have no periods, no hormonal ups and downs and no perimenopause symptoms even though I am nearly 46.
If you are thinking of a pill with body identical estradiol I would recommend Zoely above Qlaira as it is monophasic, which means it is better suited to skipping the dummy pills to remain bleed free.
The chaotic dosing of Qlaira is designed to mimic a menstrual cycle, so for someone like me taking it for menstrual suppression that would be particularly undesirable. I don't see the point of using something that can get rid of your periods yet still having withdrawal bleeds every 4 weeks. It is like having a Ferrari and only ever driving it in first gear.
You can choose any pill you like provided it is medically suitable. Not all are funded by the NHS - I have to pay for Zoely as I live in Scotland. I believe it is available on the NHS in England at least in some areas.
One thing I would pick up on though is you mentioned suffering from perimenopause symptoms whilst you are taking a birth control pill.
This is only really possible if either you are on a progestin only pill such as cerazette which contains no estrogen, or you are on a combined pill but only taking it for three weeks out of 4.
If you are on any of the standard combined pills and taking it every day ie no dummy pills or pill free week, in theory it shouldn't be possible to experience hypoestrogenism as the doses of ethinylestradiol are supraphysiological.
It may be that you are noticing reduced androgen levels - think reduced libido, loss of muscle mass and strength - which is a side effect of birth control anyway but can be more pronounced as we age, however this is still an issue on a combined pill with body identical estradiol.
I take DHEA to combat this.
However there are still good reasons to consider switching to a body identical pill, as they have a lower risk of blood clots (although this is minuscule anyway in healthy users) and contain more modern progestins that have fewer unfavorable metabolic and off-target effects.
Zoely in particular is preferred by women with hormone related mood disorders such as PMDD.