Welcome to MM.
I would put the patch on immediately, it is not a good idea to have the progesterone from the Mirena pumping out into your womb without any oestrogen in your system, my gynaecologist is dead against it. I had a Mirena coil (for contraception) when I first went into menopause but once I stopped producing enough oestrogen, it caused me all sorts of problems.
It is a matter of personal preference but having used patches myself, I would recommend Oestrogel because I think it works much better and you can adjust the dose to suit your needs. I didn't ever get enough oestrogen out of the patches, they left a horrid black mark, an itchy rash and I didn't like going around with a patch stuck to my backside.
If you do get any side effects, I very much doubt it will be on account of the oestrogen, it will more than likely be the progesterone in the coil but if you keep your oestrogen up to a reasonable level, it should combat any problems and again, this is more easily achieved with Oestrogel.
Good luck and please let us know how you get on!