The answer to this will depend a great deal on other precedents that have been set as to the definition of a fast food restaurant. It will also depend on how good the lawyer for Pease in a Pod is. This is because the facts, as stated in your question, are not clear enough for us to know whether the alcohol ban will be applied in this case.
The question at issue here is whether Pease in a Pod is a fast food restaurant. In some ways, it clearly fits the definition because it has a standardized menu and people can get food to go. The issue arises because the definition of “table service” is not clear. Pease in a Pod does not have table service in the sense of having wait staff coming out and bringing food to patrons who have ordered at their tables. However, it does have workers who circulate and refill drinks. It would be possible to construe this as “table service.” If that is table service, then the alcohol ban will not stand.
The problem is that we cannot tell what the definition of table service is. There are probably cases that have been brought in the past that have shed more light on the details of this issue. The outcome of this case would rest on such precedents to a great degree. If there is no clear-cut case in the precedents, then the outcome will depend largely on the skill of Pease in a Pod’s lawyers. If they can convince an adjudicator that personnel walking around, asking how things are and filling drinks constitutes table service, then the alcohol ban will not apply.
In short, there is simply no way to know for sure. The wording of the rule, as set down in this question, is not clear enough to tell us if the alcohol ban will apply to Pease.