Hit the hole in the sphere with a uniform beam of parallel light (e.g. a throwers hot spot at several (5-10) meters distance).
Measure the intensity just in front of the hole with a luxmeter.
(This number is lumens pr. square meter that hits into the sphere.)
Multiply with the hole's area in sq.meters and the result is the lumens number that enter the sphere - simple and easy.
The only difficulty I can see here is the sagging intensity from the thrower, when you remove the outer luxmeter and read the one build into the sphere the light might have dropped a little. Just repeat the procedure a couple of times and you will get corresponding values.