I think the notation you wrote reveals that you intended a different version of the original sentence:
It isn't raining, and if you say it is, I don't believe you.(Alternatively you would be opening yourself up to false beliefs and false assertions, such that you're saying maybe it's raining but stating that it is not. That's anomalous and I'm not sure how to analyze it.)
There is also a complication of the phrase "believe you", which is I guess some sort of grammatical metonymy where "you" is substituted for "what you said".
Regardless, this is going to involve some sort of embedding of propositions beyond what can be expressed with basic predicate logic. You would need second-order logic of some sort including potentially dealing with issues of factivity (compare "I know P" versus "I believe P", where only in the first is "P" entailed as true).
As a general approach, you can try substituting complex expressions for simpler parts, then building up the complex expression. (That still won't account for more advanced issues like quotation, though.)
Or would it need the biconditional?
Biconditionals don't generally correspond to natural language conditionals,* except in formal statements like "if and only if". Material implication refers essentially has a gap in the relevant truth conditions because it is unimportant what happens when "[if] P" is false, having no bearing on the outcome of Q one way or the other, so it is always true if P is false. That would seem to apply here as well. You are not making any sort of assertion about whether you would believe them if they said something other than that it is raining. (And again this would get into complex issues of embedded propositions if you wanted to consider the case of the other person saying it is
not raining, but regardless that is not specifically accounted for in the original statement-- maybe they are a pathological liar so if they
agreed with you, instead of believing them, you would question your own beliefs instead!)
[*In fact, no simple propositional logic corresponds to all general usage of conditionals. There has been a lot of research on this topic, and it's interesting but complex. Among other issues are conditionals of the type "There's food in the fridge if you get hungry."]