Firstly, your fiance is not at present undocumented. He is currently in possession of a Fiktionsbescheinigung, which states they have the right to be in Germany. Presumably before the Fiktionsbescheinigung and before the decision by BAMF he was given the paper ‘Aufenthaltsgestattung’. With this paper he was also legal in Germany for the duration of the asylum procedure. It is only the time before he made an asylum claim that he can be described as undocumented.
It can take time for permits to be issued but in his case it seems extremely long. A criminal conviction can be taken into account when deciding whether or not to issue a person with a residence permit. However, it would have to be a serious crime in order to prevent someone being given a permit for subsidiary protection.
Are you sure he is in possession of a Fiktionsbescheinigung and not another type of Bescheingiung while states he is to be issued with a resident permit? These are separate types of papers with different legal standings. A Fiktionsbescheiniogung simply states that someone has applied for a permit and a decision has not yet been made. The other type states that a decision has been made and the person is entitled to a permit (it just hasn’t been issued yet in card form).
Regardless of which paper he has, he can get married with it. The important issues for the marriage registration office (Standesamt) are if the person is resident in their area, if their identity is clear and if they are eligible to marry (i.e. not already married). Depending on the consequences for you in term of residency after the marriage, the Standesamt may **** into your relationship history in detail as part of the process of preventing sham marriages.