I like the relay idea. I use boat loads of relay driven power strips for projects, and man, those are handy.

Might be an easier solution - use a 12 to 24 boost converter, like this.

Normally if I need to step up power I use a constant current buck-boost, but this requires knowing the current draw for your specific LED strip. If it's a 24 volt BTF COB I just tested one of those, and it's 1.6 amps per 16 feet. But, that might not be what you are using.

This assumes of course your existing power supply can handle the additional load.