I looked at the Lee Valley calipers. It took me a few seconds to understand how they work. If I were making flutes and Lee Valley doesn't make what I need, then I would make a KISS home-brew version:
Get two thin steel rods. Use a grinder to sharpen the ends to a dull point. Bend one tip (short bend) at 90 degrees for inside and another (45 degrees?) for the outside. Drill the ends of two rectangular sticks of wood and glue the steel rods in the holes. Drill near one side of the sticks to leave room for the measuring bolt, which will go through one stick and must avoid the steel rod.
Join the other end of the sticks with a simple hinge, so you can angle the steel rod tips together and apart. It should work like a film maker's clapperboard.
Drill through the top of one stick to allow a threaded measuring bolt. Drill it near the end where the steel rods are inserted for maximum precision. Be sure that the hole will miss the steel rod. Mark through the hole to countersink and glue a washer on the inside of the other stick, opposite the hole. The end of the bolt can touch the washer when the bolt is tightened.
To use, insert the 90 degree steel rod in the flute hole and close the tool until the ends touch the inside and outside of the hole. Rotate the measuring bolt until it just touches the washer. Remove the tool and close it again to see the actual gap. The bolt will touch the washer and the tips will be the same distance as you measured before.
This is a homebrew version of the far superior Lee Valley tool. The advantage is that you can make the skinny steel rods as long as you need. It may take a little trial and error to find the most suitable steel rods that are strong enough to feel when they touch the walls of the hole, but thin enough to fit. Once done, you can use this tool on many flutes.
One concern is repeatability. The longer the inserted rods, the less precision you will get at the measuring bolt. You may want to make several tools in different lengths to optimize accuracy.