If heaven didn't forbid it, I'd sleep with her.
Based on what I read, heaven forbid is more common. But can you use its negative form?
is an idiomatic appeal for divine intervention to not allow something to happen
it should not be confused with (ending s )
which is a phrase which introduces an action that heaven has does not allow
Using your example sentence
Opposites of Heaven forbid are
An official form or pledge of God willing is So help me God
An opposite of Heaven forbids is
In your sentence, if
is the negation, then
is the original sentence.
Heaven forbid is an idiomatic expression.
Heaven don't/didn't forbid is not an idiom but you can certainly use it in the type of construction you have written.
Whether it sounds awkward or stilted is largely an opinion-based question. As a native speaker I don't consider it awkward, although I don't consider it to be common either.
Note that Heaven would only forbid very bad things, and that we don't have any good examples of things we know Heaven does actually forbid. It appears to reserve its forbidding powers for truly horrible things.
So this is saying that you think sleeping with her would be so terrible that Heaven actually forbids it, and that you would do it anyway otherwise.
I don't think that is what you want to say.
"heaven didn't forbid" is fine because it's a negative form of "heaven forbid", so people can figure out what it means from what the positive form means.