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In a situation where you run a vermin lord, bsb and 3 assassins who becomes the general? Can it be the BSB as the others can't be the general

No, it can still not be the BSB. You would need either another Lord or Hero to be your general. (page 134)

So you would be made add another character weird!

The General isn't decided when you place models on the table, it's decided during army list construction, and I think that's where you're running into problems. You simply can't "run a vermin lord, bsb and 3 assassins" as your only characters. Creating a list like that would mean that you created an illegal list, that's all.

15pts for a cheap warlock engineer solves the problem, not sure I've seen a list that didn't include a cheap throw-away warlock before xD

Your general is always the highest leadership Lord or Hero who isn't a BSB or otherwise ineligible as a general. If 2 models have the same leadership I do believe you get to pick which one is your general.