Using proportion or a diversity index when dealing with just two strains
Extended upon the suggested statistic

The proportion $p$ is definitely not a good measure of diversity. A proportion of 0 or 1 both indicates no species diversity as only one species exist. If one species, never reach a frequency of 0.5 in your data it might be reasonable though. Otherwise, something like $p(1-p)$ could be a statistic of interest.

If you're willing to use $2p(1-p)$ (instead of the above $p(1-p)$) as index of diversity, then you could make the point that it can be easily extended to $S$ species with $1-\sum_{s=1}^S p_s^2$, where $p_s$ is the frequency of the species $s$ by analogy to the genetic diversity within populations (see Nei 1973).

Note that these statistic will be sensitive to the sample size. For example, if you sampled only one individual, then this individual can only belong to one species and your measure of diversity can only be zero. You might want to compute an unbiased estimator for this statistic.

Your goal is to reduce the reader's cognitive load

Your goal is to use a statistic that will communicate easily. If your readers are going to be very much used to Simpson's or Shannon's indices, then even a mathematically simpler index might end up leading to an extra cognitive load onto the reader.

IMO, it will be hard to give a more definite answer without knowing 1) who are going to be your readers (which journal are you aiming to publish in) and 2) what are the exact data manipulation and interpretation you are aiming to do based on your results. For example, if it makes the comparisons among graphs easier by using a simpler statistic over a more complicated once, then choosing the simpler statistic is a good idea.

Good statistical analysis

In any case, it is essential that you to try out (and you did) different measures of species diversity and make sure that your conclusion is robust to these different measures and explain in your methods that the chosen statistic won't matter. That will make your point and your choice of statistic much stronger.