Sorry to hear about that happening, but I believe that you still may have a chance to get your files back. (Recovered to a different drive of course).
In usual use Recuva looks for files that have a ‘deleted’ flag, which from what you say I think may be what you had done.
However with a reformatted drive the files you are wanting to get back are the non-deleted files that were ‘live’ on the drive when it was reformatted.
Those files hadn’t been deleted, they were ‘lost’ in the reformat. So they don’t have a ‘deleted’ flag and there is a special way of recovering them.
Have a read of this (and watch the linked video), then once you have a target drive to recover to give it a go to see what it can find:
PS. With a bit of luck that .tmp file won’t have overwritten much, and maybe not any of the non-deleted file data which is what you actually want to recover.
Good luck with your recovery attempt, hope it goes well for you.
PPS. I see that you had already asked this question on reddit and got some not so helpful replies there.
Do you actually have a 2-month old backup as stated there?
And is the drive that you are attempting to recover from an SSD as some there seem to think, or is it a HDD?