No. A correct sentence would be, "He suggested that we bring an umbrella because it may rain." If the rain is in the past tense—if it is finished—why do you need an umbrella now or later? It only makes sense if you are talking about the future.

"He suggested that we bring an umbrella because it might rain."

I imagine you mean that he made a suggestion in the past that was to be carried out sometime after the suggestion was made, in either the past, present, or future. I would say "He suggested that we bring an umbrella, because it might rain." Use the present subjunctive after suggest (even to talk about the past), and use "might" (not "may") for a possibility in the past. In fact, you can you "might" to talk about possibility in the past, present, and future. You could also backshift by putting the whole thing (except "bring," that stays in present subjunctive) into past perfect, but I don't think that would be particularly useful.

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May I alos ask what is about "He suggested that we bring an umbrella because it could have rained"? Thanks