There are 3 ways of looking at this I think. 1) Indicatively 2) speculatively on culprit 3) speculatively on possible.
1) 'There is a possibility that he is the culprit.' is fine. You have 'a possibility' and 'that he is the culprit' apposed, they both name the same thing.
= It is possible that he is the culprit.
(That he is the culprit) is possible. This simply describes the bit in brackets as possible, there is no speculation, even though the word 'possible' leads you to want to use a subjunctive form.
(That he is the culprit) is certain. An adjective (certain) describes a noun phrase.
2) (That he be/could be the culprit) is possible. The speculative part of your sentence is on whether or not he is the culprit. The use of just 'be' sounds funny nowadays.
3) (That he is the culprit) be/could be possible (but we don't think so). Again, sounds funny nowadays with 'be', but is not wrong. In German it would be fine.
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