Yes, Sharp's rule applies here, and since this rule is mostly true, that is strong evidence that we should translate this as "God and Father" just as the ESV has, and interpret both nouns as referring back to Christ.
However I would argue that there are two fallacies being made in the asking of this question which make it seem like a lot is at stake in this passage when it really isn't.
Grammar has no poison pills
"The T-S-K-S construction which uses God and Father becomes a type of "poison pill" if the rule is denied because without the rule the meaning of τῷ θεῷ καὶ πατρὶ is that God and Father refers to two distinct entities God"
This assumes a rule is always true or always false. For example, if I find an example that violates subject-verb agreement in number, then it must be that subject-verbs always disagree in number! But when a rule is false, it just means that it can't be applied to resolve the ambiguity, not that the opposite of the rule can be applied to resolve the ambiguity. Otherwise we could generate lots of false rules and use them to understand any passage. Clearly a false rule does not allow us to resolve anything in either direction.
Rather, simple rules like Sharp's rule (or subject verb agreement), are at best be mostly true, and to make them precisely true you have to keep piling on more conditions or formalize irregularities (introduce different rules for collections, etc) which is why grammars are so complex and remain the subject of research even to the present day. In the case of Sharp's rule, exceptions have been found in the greek cannon outside the NT even though the rule appears to be true most of the time.
Theology does not require any grammatical rule
"Nevertheless, the unquestioned relationship between God and Father requires the validity of the rule."
This assumes that if a grammatical rule can be used to prove a theological point in one passage, then the validity of the theological point requires the validity of the rule. Yet it could be that in this case John was referring to two people without denying the doctrine of the trinity. Or it could be that John was referring to one person yet the grammatical rule is false. The doctrine of the trinity does not require any grammatical rule, nor is it established by any single passage.