Why doesn’t the sentence "the standard of proof being one based on balance of probabilities" contain a verb?

Solution 1:

There is only one sentence in your example. The text after the comma, the standard of proof being one based on balance of probabilities, contains the non-finite verb being.

Such constructions are often called absolute clauses, defined as follows by The Oxford Dictionary of English Grammar (p4):

Absolute clause: A non-finite or verbless clause containing its own subject, attached to the sentence from which it is separated by a comma, (or commas), and not introduced by a subordinator. Also called an absolute construction.

In this case, the subject of the absolute clause is the standard of proof and the sentence as a whole is grammatical.