2.5mm2 T&E can be used with a 20A breaker.
All depends on the installation method.
SWA is rated slightly higher than T&E.
Don’t like the idea of there being two 16A breakers in series, it’s a guessing game as to which would trip in the event of a fault.
As you intend installing extra sockets, and even intend using sockets for lighting, you will need to provide RCD protection.
Easiest method would be to swap the MCBs in the garage for RCBOs.