It's a bit surprising that this was booked without confirming dates with you, but I 'm guessing that they did check with your DH and he said it was OK.
Also surprising that no-one has mentioned it, either. I'm afraid bells are ringing here, DH clearly did know that you wouldn't be happy with the dates, that's why he didn't tell you, but a) how come ILs haven't mentioned it and b) why did he agree to the dates knowing that it would cause a problem?
Is there something else going on? Does he normally behave like this?
Given that you've ended up in this situation though, what can you do to make the best of it?
I reckon you could speak to MIL, and explain that you're so excited about the holiday, DH has only just told you, but the date is a problem , is there any way it could be changed? Explain you don't want to let her and DS down by cancelling ( thus wasting the money they have paid for your flight etc) but you equally don't want to let your own family down.
The other thing is, summer holidays are quite a long way off, there is plenty of time for your sibling to arrange alternative childcare for one week. I don't suppose that they would want you to miss out on your only chance of a holiday - I'd be mortified if my own brother missed an opportunity like that to deal with my childcare.
I don't think it would help anyone for your DH to go, with or without DS, but leaving you behind. It would also be very rude for you to back out if DH has agreed the date on your behalf ( even though he shouldn't have done). It's the sort of thing that creates long term family rifts.
Was there any discussion about dates when ILs first made the offer, or was it all just a general thing at that stage? Did you mention your long-standing arrangement to the ILs?
It does sound as if there is more to this.
But I think you have to look at how to deal with the situation you are in and how to move forward, I don't know that there's much to be gained from saying who is right or wrong, unless you are taking the view that DH has done this deliberately to upset you, in which case deciding whether to go on holiday is the least of your problems.