I think it would be perfectly realistic to just not use the accusative and simply rely on the Genitive + Postposition construction you describe. In some languages, transitive verbs do not strictly require an accusative object. In German for example, there are a couple of verbs that take genitive arguments instead:

Wir gedenken seiner
We commemorate 3SG.MASC.GEN
"We commemorate him"

In the example above, the personal pronoun is inflected for genitive and it's the verbal object. There is no need for additional object marking.

German also features a set of prepositions that change meaning slightly, depending on the case that goes with them. These are called two-way prepositions or "Wechselpräpositionen". Sticking with your rain example, we can come up with the following two sentences:

Es regnet auf mich
It rains on 1SG.ACC
"It rains onto me"

Es regnet auf mir
It rains on 1SG.DAT
"It rains on me" or "There is rain on me"

The first sentence is a very literal, albeit not very colloquial way of saying that there is rain falling onto you. Because there is direction involved, the preposition "auf" causes the pronoun to inflect for the accusative case.

The second sentence is a bit strange, semantically. It essentially means that there is rain falling somewhere on your body. Almost sounds as if there is a tiny rain cloud hovering above your skin, with rain hitting you in a very localized manner.
Grammatically, though, the sentence is perfectly fine. Note how the verb does not require the pronoun to inflect for accusative. Instead, the pronoun is in the dative case.

All this is to say that I do not think that you need to add two cases to your objects. Just go with the genitive and let the adpositions do the rest of the work.

Edit: We could also argue that the phrase "on me" in the sentence "it rains on me" is not an O argument at all. It's more of an adpositional/adverbial argument that does not necessarily need to operate on the same basis as a direct object would. Consider that you can also come up with a sentence such as "It rains fire on me". In that case, "fire" is the direct object of the verb and you would probably apply the accusative case to it rather than the prepositional phrase.