Omitting "that is/that are" and "its/their"

The two works are similar and it is not just because they are both from franchises (that are) notorious for (their) poorly-written characters.

It is perfectly acceptable to omit the "that are" and "their" in your first sentence, as they are attached to relative clauses, and match the case when their omission is optional.

The two works are similar and it is not just because they are both from franchises (that are) notorious.

The use of "notorious" as a postpositive adjective in this case does indeed sound unnatural.

The two franchises are notorious for (their) badness.

It is perfectly fine to omit the "their" in the second sentence, depending on the importance of specificity of the "badness."