The only adhesive that is archival is to use rice paste and mulberry paper for hinges. ALL of the other stuff out there that has a chemical adherent and a chemical carrier which is what sprays and ATGs are, are not archival!!!
No matter what the manufacturer is trying to tell you the chances are that they're only stating that their adhesive is PH neutral and passing that off as archival, which is a far cry from actually being archival. For putting the mats into place, the ATG is fine as you have said they aren't for longevity.
As the pieces aren't going into a frame, then you will need to stick the mat down instead of hinging it.
Try to avoid spray adhesives as that is a messy way to do it. It will get on everything and possibly onto the front of the image whether you want it or not.
Actually, a better way for you is a product out there that is a piece of foamcore that has an adhesive already on it and a cover sheet to protect it. Thisway, you can cut a piece of this fc to a size larger than the overall size of the mat, stick the photo down and burnish it in place, then just stick the mat down, and then trim the fc to the edge of the mat and you are done!!
Quick and easier than sprays and such.
