We went out as a four (my BF his DS, me and my DS) about 2m after we met, but we were 'friends' at that time, we'd only kissed once I think 
It was a rip-roaring success!
The boys get on better with each other than with anyone else they've met, they bring out the best in each other and for all of us to be together is mostly easy. We spend most saturday's together as a four, and sunday night is date night, it's working, so far so good!
Them getting on, and us getting on with each others boys does really help everything go well.
In an ideal world, delaying meeting up with the DC is ideal, as it gives you valuable adult time/date time, and protects the DC a little. But when both partners are FT working parents, 1 partner has their DS 100% of the time, and the other one 50% of the time, it's flipping hard to find any time at all , so it makes sense, if you are serious about committing to a relationship to try to make it asw normal and accepted as possible in a reasonably sensitive timescale. It depends on the DC as much as anything, our DSs are 5 and almost 7. If they were younger it may have changed things a little.
I do feel that it's highly appropriate for our respective boys to see us happy together, that it's OK for us to hold hands, hug, light cuddles etc, as that's what a normal relationship is all about. For one reason or another, both of us have had difficult relationships with the other parent of our children, those children need to see that what is normal is different to what they saw, they need to grow, heal and learn from a healthy environment.
Each situation is different, communication and understanding is key, fwiw OP, I think you should accept what your BF says for now, and this situation may change in time, give HIM the chance to relax. He's doing the right thing by his DD somehow, but he may adjust that timescale as your relationship develops.