I'm sure I've posted some of these thoughts before but here goes anyway.
I've got copies of old Acorn User magazines (83ish) advertising the BBC A for £299. Far bigger PCB and case than the Elk, plus stuff like the teletext chip should have provided possibilities for cost cutting. Design out all of the PCB and sockets / pads that aren't required (have the same standard ports as the model A), get a smaller case, perhaps a chiclet keyboard like the PCjr or spectrum+, with perhaps an enhanced ULA to replace some board logic. Admittedly upgrading the RAM to 32k would have had a cost.
As well as the hardware over the years I've heard rumours about the financial side of things. Did Acorn pay the BBC to use the brand name which I'm sure read/heard that they did? Did Acorn actually offer schools discounts and make a profit, I've heard figures between 20-50% mentioned over the years. If so it might well have been possible to cost engineer the price down of a stripped down Acorn electron badged BBC A (with 32k) to the magical £200 mark.
Another interesting question is did Acorn pass on the design and development costs of the ULA directly onto the Elk buyer or did they just spread the cost as overheads?
We all know the delay hurt the Elk badly and made a real issue out of price and spec (esp. the amount of RAM) because fundamentally the Elk was a cut down version of a 1981 computer (albeit a very good computer). The Spectrum 48 by 1983 was retailing at just below £130 and with the Elk listed at £200 in '83 if you could get one - the Elk had one chance and missed it. By mid-1984 the C64 was retailing at around £200. In mid-1984 the CPC 464* came out, £250 for computer with built-in tape deck and a mono monitor. The Boots advert from Christmas 1984 (courtesy of @russty_russ on twitter) is quite sobering, the competition it was facing was tough.
At the time of that advert in late-Nov early-Dec the PCN sales charts of the time were C64, Spectrum (not including the new Spectrum+), CPC (colour?) and the Elk. Then you'd have the beeb, spectrum+, C16, MSX etc. Just six months later in the last issue of PCN it was Spectrum, C64, CPC, Beeb, Atari 800XL, Elk.
* Amstrad had ULA problems with Ferranti as well, legend has it that Alan Sugar picked up the phone and gave one of bosses a right earbashing and eventually LSI Logic made the chip for them.