Why?
Assuming GCN similar to todays, AMD should definitely be able to cram in tahiti amounts of shader cores + their little x86 cores in under 200mm2 on Samsungs 14nm process, either LPP or the upcoming more cost optimized version. Lets assume $7000/wafer (It will likely have dropped by then, both due to maturity, and as their biggest early process adopter(s) will be looking to move on to 10nm). 200mm2 die area fits 300 dies to a wafer. Assume worse than real life yields - only 2/3 of the dies are OK => $35/die add costs for test and packaging et cetera. Lets ballpark cost per useful APU at $50. (This excludes development and licensing costs). Next major item will be memory. By then 8Gb GDDR5 will have been produced for two or three years by Micron, Samsung and Hynix. Would be surprising if that would cost more than (ballpark again of course) $50 (maybe a single stack of HBM could be competitive overall). Ditch optical drive, (distribute on Flash), ditch hard drive (use inexpensive Flash-drive, expand via USB3 if customer so needs) add $40 or less for 128GB or possibly 256GB, again distribution off Flash reduces need for built in memory, and this sucker needs to be cheap! Add networking and other small bits and bobs, power supply, cooling, controller and packaging. We should be at roughly $200 in total. Add retailer margins, and whatever I may have forgotten, and sell to consumer for $249. Tadaa! Side benefit - no moving parts except for fan.
So - yes I definitely believe Nintendo could launch a 4TF console with a Bill Of Materials well below $300. Development costs have to be accounted for somewhere, and won't be trivial, but that's normal.
I don't believe they will do this, much to the chagrin of most of us here. But that's not because they couldn't, the core of this device simply requires them to talk to their old partner AMD, and say "We want these specs, at this point in time", and then haggle for a bit.