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There is nothing usual or unusual, it is just a different situation from others.
So, if I understand correctly, you want to move to one place in Japan and asked the company to make this possible with continued remote work in your current job, which they are not obliged to. They denied, but offered a placement in their factory in another Japanese city instead. This is, from their standpoint (and mine) a good compromise. Now it is up to you to decide whether you accept the disadvantages (from your standpoint) of this compromise, or resign and look for a better fitting job.
Also keep in mind that expensive international schools are a luxury, as far cheaper (local, public) schools are available, again with some advantages and disadvantages - like all choices in life.
At least you have now the wonderful starting point of clearly knowing the options and their circumstances, so you do not have to decide based on incomplete information and guesswork (which is far more common).

Hi,

Thanks for your reply.

Its sort of what you said, but not quite.
I’ll still be doing the same job I do now, just with a presence on site in Japan more frequently than I planned. It means that while I continue to do my job (which is possible 100% remotely), I’ll be able to act as a physical contact point between the two sites.
As local education is all in Japanese, that isn’t really an option, as it may disturb the continuity in her education when moving back to the UK or onto another country.

From my standpoint my place of work is still the office in the UK, so it doesn’t seem unreasonable to ask them to cover travel expenses to the office. But there are definitely different ways of looking at it and I’m obviously biased towards saving £7000 a year on train fares.

My advice above is not changed by this new information.
Of course if you are planning to stay only for a year or two, switching your kid's education to Japanese would seem disruptive. But if it is for longer: Additional language skills (and giving her the chance to not be an outsider) are an asset not to be underestimated!
It is up to you what you decide now, but in my opinion money should not be the main factor.

I've worked as an expat several times, and totally agree with Beppi.

There's nothing in that advice, or in mine, that precludes further negotiation, however.  I recommend coming up with an economic case for your employer to pay you more after the move. 

Your personal desires, after all, are not a business reason for your company to spend more on your move than you cost them at home.  Your current job already has a monetary value to them, which tends to put something of a ceiling on you.  The solution may be to move the ceiling.

From your description, you won't exactly be doing "the same job" in Japan:  you'll also be doing a lot of liaison work and coordination that can best be done on the scene, that should be appreciated in both the UK and Japan, and that has a monetary value.  In your place, I'd quantify that, make it the center of my business case, and reopen discussions.  You might be pleased with the response; presented diplomatically, at least it shouldn't hurt you.

Good luck!