Would I be eligible or the S2 or would I have to go down a different route?

S2 type isn't ok because you both haven't been married. It requires your marriage certifcate and I am afraid that you can't provide this. Since you just stay one month or two, you can just apply for a L type with your BF's invitation letter and copies of his passport information page and resident permit. If you just got a duration of 30 days, you could extend it for another month from local entry and exit management bureau at least 7 days before it expires.