In describing your situation, he's called the cops—but the show hasn't begun yet.
It sounds like you want to express two different things:
- He called the cops.
- The show hasn't begun yet.
The first is in the past tense and the second is in the present tense.
You are correct that this is ungrammatical:
✘ He called the cops on us before the show begins.
This would be correct, except that it puts the first verb into the present tense:
❔ He calls the cops on us before the show begins.
It is grammatical, but it doesn't convey the sense you want.
This also would be correct, except that it leaves the meaning of the sentence ambiguous:
❔ He called the cops on us before the show began.
It is also grammatical, but it's not clear if the show has yet to start, or if it's already started, when the sentence is uttered. So, it's not what you want either.
In order to make your meaning clear, you're going to have to do it by turning it into two clauses with a conjunction:
✔ He called the cops on us, (and / but) the show hasn't started yet.
This is grammatical, natural, and conveys the meaning you want—even though it loses some of the simplicity of any of the single-clause sentences. It's something that might be said to a third party about events that have transpired. (Note that it's more natural to use started here than begun.)