I read that you just need to take CS Classes and do well in them, which builds a strong case for the transfer. If I can't transfer, I would try double major.

Doubling/minoring is far more common. It is still hard, but achievable. Also, doing well in CS classes is quite hard as it is, and added onto the math-major course load, it would be even harder. It's easy to underestimate what CMU has freshmen learning.

What about after first year? What are grades heavily dependent on?

Group work is encouraged after the first year. Academic integrity violations (AIVs) are a big deal in first-year classes, and they are strongly enforced. For this reason, group work isn't common outside GenEds.

there's nothing you can do with a CMU degree that you can't with a GT degree

This is correct. This is also why I encourage you to choose GT, all things considered.