Using bigger needles for the cast on only works for cast-ons that are made with a single strand (like knitted cast on or cable cast on), not for any long tail cast on. It's the tail strand that's too tight on long tail cast ons, and using a bigger needle does nothing to correct this.
The twisted German cast-on is excellent because it has an extra twist in the tail yarn, which automatically makes it add more tail length in the cast on.
But what makes it even better is if you cast on and leave some space between cast-on stitches as you tighten them up. If you leave a space between each cast on stitch and the last one that's about enough space for an extra stitch to fit on the needle, it will use extra length of the tail yarn and make the cast on much stretchier.
You'll need almost twice the length of tail yarn for the combination of the twisted German cast on and the extra space between as you will for a standard long tail cast on. Be sure to allow a lot of tail length.