I haven't built one of these myself, but one of the best resources I know of for the theory is antenna-theory. The second page mostly discusses gain. There it talks about optimizing the distance between the feed antenna and the dish, given a fixed pattern for the feed antenna, but basically the same thing applies to narrowing or widening the pattern of the feed antenna for a fixed distance.
There isn't a lot of worked out math there, but it discusses what the tradeoff is. The more obvious component is "spillover loss" — any part of the feed antenna pattern that doesn't hit the dish is wasted power on transmit (and a source of noise on receive), so you want to minimize that. The slightly less obvious component is "taper loss" — the dish is most efficient when you're using all of it. If you concentrate 99% of your pattern within half of the diameter of the dish, you might as well just use a dish half as big (with accordingly lower gain).
We can work backwards from this graph which shows the efficiency of different values of the angular size of the dish given a pattern that's shaped like $\cos^{n} \theta$, for different values of n. The trick is to figure out what half-power beamwidths those ns correspond to, which is a simple matter of solving $\cos^{n} \theta = 0.5$, or $\theta = \arccos \sqrt[n]{0.5}$. This works out to
$$
\begin{align}
n = 2 && \theta_{HP} = 45^\circ && \theta_{maxEff} = 67^\circ\\
n = 4 && \theta_{HP} = 32.76^\circ && \theta_{maxEff} = 52^\circ \\
n = 6 && \theta_{HP} = 27^\circ && \theta_{maxEff} = 45^\circ \\
n = 8 && \theta_{HP} = 23.5^\circ && \theta_{maxEff} = 40^\circ
\end{align}
$$
where $\theta_{HP}$ is really a half-power half-beamwidth, and $\theta_{maxEff}$ is me eyeballing the peak efficiency values off of that graph.
This indicates that your guess was not terrible, but it loses a little too much power off the edges of the dish. The ideal seems to a half-power beamwidth between 55% of the angular diameter of the dish (for feeds fairly far from the dish) to two-thirds the angular diameter of the dish (for feeds fairly close). Having it be 100% of the dish diameter is 1-2 dB worse.
