I was on the receiving end of an attitude like yours - it's incredibly entitled and you clearly don't like your BIL much so I'm not surprised he's not entertaining giving your child an opportunity, even if it's within his gift.
I worked for a large media organisation and was asked from time to time if I could host work experience for the children of friends/relatives. Initially I was able to help with some of them but then, quite rightly, the whole work experience system was taken over by HR and had to go through them. It was somewhat of a relief as some of the teens were frankly embarrassing in their attitude to an opportunity that others would have given their right arm for and I felt it was more about the parent being able to boast about where their child was doing work experience, rather than the child actually wanting to do it there.
Do your child a favour and let them organise it as it will be much more meaningful for them. My own teens sought out their own, getting a few knock backs along the way, and both now have paid holiday jobs at the same organisations.