What I am trying "prove" is to create a pretty solid argument that using this approach is more robust and useful than using transformed data and a linear model based ANOVA. This would add a level of confidence in the differences we see on the graph. There will be concrete differences between the lines.

I feel that you may be setting yourself a very difficult if not impossible task. All we have done so far is go through the mechanics of how to find a SE in an unusual situation, and use it to get some p values. There is no discussion of robustness, nor of goodness of fit, nor of usefulness. It is a brave (read risky) matter to go against a supervisor, and in this case I think you would need a much deeper and more solid foundation in statistical theory to prove your point, if indeed it is true. (There are other simpler models you could consider as well.)

As for the arcsine transformation and the anova, this isn't really an anova situation at all as far as I can see. Anova compares the means of different groups and sometimes the data that gives those means needs to be transformed so that the mathematical anova calculations will be valid.

On the other hand, insects are not my field. Your supervisor is a better source of advice than a random guy of the internet. If you are really concerned, consider buying an hour of a statisticians time where you can discuss the whole thing face to face.

Good luck, kat