I think it is a nice idea, provided that you find a crt portable TV with enough space inside. Many of older portable tvs have a huge space for the batteries that you can remove, and a big and bulky transformer that, in theory, shouldn't be so difficult to replace with something more compact.
Be aware that you need to built in the unit also a sensor bar and that you should play from 30-90 centimeters from the screen. There is a possibility that a small screen (let's say a 5'' that is the most common screen for portable color crt Tvs) is not sharp enough to be readable and that playing wii games from 30-50 centimeters is not really enjoyable.
But a wii has a lot of other advantages over a gamecube: usb and sd support, wifi and horsepower for some interesting emulators or mediaplayer or whatever and a real rgb output that should provide the sharpest image on the CRT. The gamecube NTSC doesn't have rgb out (but pal units have it).
To check if the TV of your choice has a decent sharpness you can try to do some trial with a raspberry pi and retropie.