The third party makers seem able to make various gear hubs fit with little difficulty.
Brompton, with its maker's research and development facility ought to be able to do it at a canter.
You completely miss the point. Of course Brompton could do that. But it does neither fit what they want to achieve nor the customer demand. Plus you cannot compare the way how small one-man-bands work with big scale production. Brompton produce around 50.000 bikes a year. The multi-hub-gear modifications are mostly done by Juliane Neuss and by Ben Cooper. Steve Parry seems to have lost his prominent spot (and maybe interest as well) in modifying Bromptons, Vostok (who offered a wider rear frame) seem something between very unreliable and out of production, Graham of Tillercycles gave up his business. Other than that there is Brommiplus, who sell S/A 5- and 8-speed kits from Taiwan and Vincend van Eerd, who creates very innovative builds but also not at big scale. So the only ones doing something like a manufacture production seem to be Juliane and Ben. I know Juliane pretty well and thus know about how many conversions she has done over the last ~20 years since she offers it (first Nexus 7, then Nexus 8). Ben seems to post most of his builds on instagram or his website. It is a pretty safe bet to assume that
over the last ~20 years not more than maybe ~2500 of those conversions have been made overall, including all offerings (and that is a
very optimistic guesstimate). Again: Brompton built 50.000 bikes
each year (and have sold more than 600.000 bikes over the last 20 years, the period when those max. 2.500 conversions were made). While this is a nice business for mechanics aftermarket (and good for the customers that it exists) it is simply not a business case for Brompton, even considering that probably many more people would buy a multi-geared hub if it was offered directly from the factory.
I have little sympathy with Brompton over costs.
They've been making a £300 bike and charging a grand for it for years.
This is ridiculous and you know that. If the bike is that bad and overpriced and does not even offer what you want you are simple not their customer or the customer they are targeting. The big question is: Why do you own a Brompton then, if it is so bad? Why don't you have a differnt bike? And if a multi geared hub is so important to you and you still want a Brompton (whyever you want to have this
overpriced bit of outdated technological crap 
): Why didn't you go for one of the many conversions that are available? Juliane offers her conversions for more than 20 years now, and the same goes for Steve Parry and Ben Cooper. Maybe you are demandeing something that you are not willing to pay for? So possibly it is not so essential to you, despite you are claiming the opposite...
Part of the problem is current management seem more interested in special editions based around fad and fashion, and 'trendy' Brompton Junction stores than they are in improving the product.
Brompton are running a business and the job of the management is make this business survive successfully. They have done exactly that. They are constantly at the limit of their production capacity and have been for years. So obviously they do something right, judging from the market. Just not for you - but how many Bromptons have you bought and how many will you buy over the following years? So how relevant ist you opinion?
Don't get me wrong: I heavily dislike Brompton's current way of becoming a fashion item and leaving the way of being an engineer's product. I hate that they drop their USP of sustainability by no longer offering spare parts for older bikes. I hate that in some aspects the quality of their accessories (like bags) seems to become lower than in the earlier years as they look more for fashion than for functionality while the price is still going up. I hate it that, after 30 years, they cancelled the contract with the German importer and plan to sell directly trough their own distribution to the German dealers (as they have done before in the US and Benelux and elsewhere). This will make many things way worse for German dealers and customers. They are following a growth strategy and that means marketing, centralising and cost reduction and this means (upon other things): less variability (as you can i.e. see on the brakes and the suspension block) and ditching non-lucrative but expensive elements of their business (like spare parts for older models).
Given their hamfisted attempt at an e-Brompton, perhaps they are better off staying in the 1970s and waiting for the axe to fall.
I cannot get the hate that some people, especially in the UK, seem to have against Brompton. The Brompton Electric is, compared to it's competitors, a good bike and not even overpriced. And it is absolutely mandatory to have one for a bike company if they want to survive the next couple of years. It may have it's issues and it is expensive but it gets updated regularly softwarevise and in comparison to other offerings (may it be other electric conversion kits for the Brompton or other pedelecs from major quality brands) it is not overpriced.
Just go and buy a better folding bike. Interested if you are able to find one. If you are not and you are so dead sure that it would be easy to perform better than Brompton does: Go and create your own brand.