This is interesting. It seems that the Likud is putting forth efforts to include exceptions to the minimarket law. The Likud would like to exclude the city of Eilat from the limitations of the minimarket law, meaning to allow businesses in Eilat to operate on Shabbos. As well, they would exclude quick stop shops in gas stations.

Likud made efforts to vote on the exceptions, but canceled the vote when they realized they could not get a majority - Yesh Atid agreed to support some of them, but Labor and Meretz did not. As well. there is internal debate on UTJ and Shas on the matter.  The Likud was willing to grant the exception to the limited area of the area of the hotels in Eilat, but it did not matter because the Gerrer Rebbe told MK Litzman to not agree to any exceptions, even just the hotel area.
source: Walla News

I agree that Eilat should not be excluded. Eilat is dependent on tourism business, but many cities would reap similar benefits and will be hurt by not allowing the stores to open. If Shabbos is Shabbos, and business is not being allowed on a national level, why should one city be any different from the other?

That being said, I can see a loophole being found for Eilat to allow businesses there to operate on Shabbos, should they so desire. Many rabbis hold and have held, Rabbi Elyashiv included, that Eilat does not have the halachic status of Eretz Yisrael. If Eilat is not halachically part of the Holy Land, perhaps it is not an affront to the Jewish character of the State to allow them to operate on Shabbos. Maybe that can be used as a loophole. It might ot be a good idea to have such an exception, but I can see it being possible.