OP - I totally get where you are coming from. You are in a bit of a lose-lose situation which is unfortunate.
I have friends who have excluded children from weddings 'cos it would up the numbers (and therefore costs) massively but that logic is a bit negated in this particular case as you are allowed a plus one.
However, even if you explain nicely and even if your friend totally understands she will still probably not be able to change it as she would then have various other guests getting upset cos they couldn't bring their children but you could.
So a lot depends on how many other guests' children are being excluded and if she was even able to make an exception for you.
Even brides can be caught between what they want and what is practical it isn't necessarily her being precious about naughty children disrupting etc.
I was able to invite all children to our wedding, but that said, not many of our friends had any so the difference it made to catering etc was negligible - but everyones guest list profile and budget differs so it can be a complex decision - not just her being anti-children.
Your situation (long way to travel, low finances, close child etc) is probably unique amongst her guests so something a blanket rule, which this is, doesn't take into account.
I would have a nice pleasant chat to her about how you are thrilled for her, explain your position (you did it very well in OP) and then even if she can't alter it - and you can make it clear you understand if that is the case - at least she will know you didn't go out of pique, but because it just didn't stack up - sadly.
Or can you get creative? Do you have a mum who might fancy a little trip away who could go with you - you attend the wedding she entertains your ds - then you go back to your accommodation to meet up with them and you all get a little break away?
Could you go away from one night only? He might cope with your absence better than you think? Sometimes our dc's can surprise us?
good luck.