"then giving it up is also going to have consequences for the OP"
I don't for a minute think she should give up on her studying, what she's worked hard to achieve and her dreams for a future job.
The offer of this course means there is a lot of thinking to be done to figure out whether to take it, and if so, under what arrangements, and if not, to what end.
So the first thing to be really clear about is what is the exact purpose behind pursuing this course of study.
The next is whether that purpose can be equally, or better, met by doing another course. This will involve putting some thought into where the family are likely to end up, and when.
I would say next thing to consider is what the options are for taking the course - how long would she be away, should the children stay put or come with her (it sounds like some of these decisions have been made).
Next you need to think through ALL the various impacts the absence of a parent, for a year in another timezone, will have. How will the children cope (not children in general, THESE children)? What strain will it put on the marital relationship? How will it affect family dynamics? (Again, THIS family, THIS marriage - it doesn't really matter nobody in the 3rd world ever spent a full night under the same roof as their spouse.)
Then you decide if these possible impacts are worth it, and work out ways to minimise the negative impacts.
Finally (and importantly) what will make the you happiest? How much will you miss the children? Will that hurt impact on your ability to study? Will having your qualification make up for feeling estranged from your family when you return? Basically what I asked earlier - will doing this course make you happier than being away from your children will make you sad? Not an easy question to answer, but a crucial one.
Finally, don't fall into this kind of fallacious thinking:
"If the OP has been offered a fully-funded postgrad at a good university, then the chances are she's pretty smart and gifted. These kinds of opportunities are pretty competitive and she'd be VU to pass it up."
You can pass up any opportunity if it's not the right opportunity at the right time. The problem with things that other people see as rare chances that you must not turn down, is that it's very hard to judge them clearly.
If you're that smart you'll find another way to get where you want to go even if you don't do this course. I really think that you need to be a bit clearer in your head about exactly where you want to go before you can make the right choice.
You have a lot of thinking to do OP. Good luck :)