Interchangable compatablity of circuit breaker brands

Short version: you can't cross breaker brands. You cannot put a competitor breaker type in this panel.

A "tandem" breaker is not what you want. Here is an article about those.

What you're hoping to use is a THQP breaker, which appears very similar to a tandem breaker, however, is actually a 2-pole breaker that is half-height. On certain GE panels, GE has a system that uses 1/2" tall (half-height) breakers, 2 per breaker space. Their 2-pole breakers in that system are 1" tall and straddle 2 breaker spaces, with a 1/2" breaker above and below them.

You can't use them. Those half-width breakers clip onto a special cross-bars added to the bus stabs, and your bus stabs don't have them. You can see the cross-bars in the lower 5 rows of this panel here: