First of all, thanks to all of you for your suggestions on how to handle the situation.
Yesterday after he got finished with work, my lovely and level-headed boyfriend (R) went back into the store to tell them that their Guest Relations department basically told us that we were up a creek. I did not go with him, because I did not want to end up as a post on customers_suck.
It came down to this:
1.) Store Manager takes one look at the receipt and sees that the transaction was rung on a register they've been having trouble with.
2.) SM apologises that we were given the run-around and were passed from one person to another.
3.) SM loads $25 on the gift card and apologises again. She hopes we will continue shopping with Target.
Ten minutes. Ten freaking minutes. All it would have taken to resolve the issue last week was for Rep 4 to contact the store like she said she had been doing all along. On the other hand, had we spoken to the store manager ourselves (she wasn't available when we went in the first and second times), we could have had it fixed without ever having to involve GR, but that's neither here nor there at this point. When I couldn't get results at the store or through the 800 number, I felt my only choice was to email. Now I know better. :)
Again, I would like to say that despite not knowing what the issue was or how to go about solving it, other than to refer us to Corporate, the store staff had always been very polite and apologetic. The fault was not with them, but with a wonky register and a string of GR reps who seemed to be more willing to write us off as possible scammers than to actually help solve the problem.