In reality, pretty much all adhesives affect the paper you are sticking down. The solvents in the glue migrate into the paper and sometimes will actually change the chemicals in the image on the front.
This usually takes quite a while to happen but there are some adhesives out there that will mess it up quickly.
As you are selling gift cards that will not become family heirlooms, you would be pretty safe using a glue stick. The problem I see with any method you use is that unless you get the stuff right out to the edge of the paper, you will have corners and edges sticking slightly away from the card, possibly catching on something and peeling off or bending a corner.
Have you looked into cards out there that have a pocket for the photo to go into?
When I made my own cards (eons ago!!) I had the card printed with a nice line around where the photo would go and while doing that the printer also punched a diagonal slit in the card for the corners of the photo to go into. This made it seem like they were buying a small print rather than just a card. No adhesives there.
Another possible method is to use ATG tape. It is usually dispensed from a gun from the roll. The paper carrier is removed by the gun and all you have on the back of the photo is a strip of adhesive. This might be even easier for you as you will be doing quantities of cards at a time. Run a strip on all four sides on the back of the photo and stick it down. You better know exactly where the photo is to go as you really can't reposition this stuff.
Neither of these methods are archival but then if you want archival, you would have to charge ALOT more for them then you could get.
Don't know if this has helped but, there you go.