What do you actually mean by a court order? You mean that his ex and he couldn't agree?
TBH, I think that's a red flag. Before I get jumped on, yes there are times when it is necessary and one person is not at fault. But it's enough to raise an antennae for me, if two parents don't agree access without a court stepping in.
My friend has two kids and as they've got older, she and her ex have agreed some time when they have one child each - it's nice for the kids to have 1:1 time and I think that can be harder in a one parent family who doesn't have a full 7 days a week to make that happen in.
But to exclusively see only one child at a time and for it to be court ordered...
You know, dating is getting to know someone. Although you don't get to be the Spanish Inquisition, something that fundamental is an important part of their life - I can't imagine not asking, just as I'd want to get to know them.
If they were too private (as PP said - secretive?) they just wouldn't be for me.
How long have you been seeing him that you don't feel comfortable asking?
Sounds like you have the gut feeling that it won't reflect well on him.