You actually make reference to something which is of crucial importance to the answer to this question:
"With a tokamak, I imagine that if you double the linear dimensions, the plasma volume (and hence the power production) will increase eightfold, whereas area that you have to protect against fast neutrons will only quadruple. So once you master the tokamak technology, you would only need to scale it up appropriately to bring down capital costs."
You suggest that the fusion power of the tokamak scales roughly as $\sim R^{3}$ (where $R$ is the tokamak major radius), but the surface area inside the device onto which the fusion neutrons are incident only scales as $\sim R^{2}$.
This is fairly accurate, although the scaling of the fusion power is closer to $R^4$ or $R^5$, for reasons I will mention later. However, from the context of your remark it sounds like you're implying that this difference in scaling would make it advantageous to scale tokamaks up to an arbitrarily large size. The reality is quite the opposite in fact.
The fact that the internal surface area of the tokamak scales less aggressively with $R$ than the fusion power is quite possibly the most fundamental reason that we cannot build very large tokamaks. This is because the neutron flux on the inner wall of the device scales like the fusion power divided by the surface area, so roughly as $\sim R^2$.
The materials which line the inner wall of a tokamak can only withstand a particular fluence of fusion neutrons before they must be replaced, as the neutrons cause significant structural weakening. Replacing these components is an extremely time consuming and expensive affair, as it must be carried out entirely by remotely controlled robots due to unsafe levels of radioactivity inside the device. The interior wall of the JET tokamak was replaced recently and the project took over a year to complete.
So the the length of time for which you could run a tokamak fusion power plant before a major shutdown is required to replace the wall scales as $\sim R^{-2}$. Clearly this is a serious problem for a very large tokamak, as the inner wall will last an unfeasibly short time, making economically viable electricity generation impossible. After all, the goal of fusion energy research is to solve the looming energy crisis, so we must be able to produce electricity at a cost at least comparable to other renewable sources or there really isn't much point building a reactor in the first place!
Although this thread has been inactive for quite a while, you actually had the answer in the question without realising it so I felt the need to let you know!
A small aside: I mentioned earlier that the fusion power scales more like $R^{4}$. This is because a larger tokamak has a greater distance between the centre of the plasma and the wall, and this allows a higher core plasma pressure to be achieved. This in turn increases the fusion reaction rate, so as you increase $R$ not only do you have a greater plasma volume, but you're also getting more fusion power per unit volume.
