Sounds like you dont need any mounting at all. Just place the image and mat board in a 5x7 clearbag ( which will allow people to view and handle them without leaving finger prints on the images.

If I am confused about what you are doing then you can also consider v-hinging with a water activated reversible hinging tape such as Lineco Gummed Linen Hinging Tape.

A v-hinge is a variation of the t-hinge that allows float mounting an image to a backing board. Detail towards the bottom of this page: hinging image to mount board