The two-way fence
You fence a competitor. Within a month, they fence you back.
This is the normal outcome, not the failure case. Geo-fencing is cheap to copy, so any street with two good operators on it ends up with both messages landing on the same pavement. Planning for that from the start is what separates a fence that keeps working from one that gets neutralised in a quarter.
What settles it is not who bids higher. Neither of you is bidding. It is which message is worth more to the person holding the phone, and a member who already has a balance with you is holding something the other store cannot match on the spot. That is why the copy points at the card rather than at the competitor: a discount they would have to sign up for is a worse offer than points they already own.
The tell that you are doing it wrong is a message that only makes sense if the reader knows who you are competing with. Write it so it would still read well if it arrived while they were standing in your own car park.
Point at what the member already holds, not at what the other store lacks.