In that case, letting the hen hatch & raise a single chick is definitely the easiest thing to do.
If you just want to know whether he carries the blue egg gene, and how many copies, there is a genetic test:
https://orders.iqbirdtesting.com/product/blue-egg-gene-with-feathers-sample/
The test is definitely faster and probably cheaper than raising chicks to find out what color eggs his daughters lay. (Although it might not be as much fun as raising some cute chicks!)
But if you want to know what genes he carries for brown eggshells (make a blue eggshell into a green one), then test-mating is the only way to go.