That is a massive crock and you are getting screwed or at least being given bad info. I'd bet the person you talked to had no idea what they were talking about. I just got back from Target and they price matched down to $159.99 and we walked out with the $20 GC. No fuss at all. Took no more than 5 minutes, if that. We had the local BB ad (stores are very close to each other) and they didn't even check BB stock even though I knew BB was fully stocked.
I understand that this can be a YMMV type of situation but Target's policy is that of doing a PM whenever possible. If I may suggest heading down to the store and being super nice to the folks in the electronics department you are very likely to be plesantly surprised and walk out with a shiny new 3DS XL and possibly a $20 gift card as well.
Actually, the policy is that the price match does not include gift cards or BOGO offers. Once you price match the item, any gift card promotion should be void once the price is lowered. The people who are getting the price match and the $20 gift card are lucky that the store isn't following the policy to a T.
If you guys successfully price matched with the gift card, awesome. However, that isn't technically suppose to happen so do not get mad at the employees if they deny it. Some people may be getting in trouble for allowing that to happen. It is definitely YMMV. It depends on how strict the store manager and district follows policy. The stores in my area will not give the gift card too.