When you have a set of categorical variables, you can calculate Gower's distances between objects and then use common distance-based indices of functional richness, evenness and divergence in the FD package.

I think the categories will themselves be the functional diversities. Chi squared tests are the best tests for categorical data, so you don't need to transform the data since the test works with non-parametric data. Don't worry about functional redundancy, since it is just an inherent property of the ecosystem. There are no rules in ecology, or analysis, so just do what feels right hehe :)

For categorical variables, functional diversity indices are often calculated based on the variety of categories and their distribution within a sample. Two commonly used indices are:Simpson's Diversity Index: [ D = 1 - \sum(p_i^2) ] where ( p_i ) is the proportion of individuals belonging to the ( i )th category. This index ranges from 0 to 1, where 0 indicates low diversity (dominated by one category) and 1 indicates high diversity (even distribution among categories).Shannon's Diversity Index: [ H = -\sum(p_i \cdot \ln(p_i)) ] Similar to Simpson's index, ( p_i ) represents the proportion of individuals in the ( i )th category. This index provides a measure of uncertainty or information content and ranges from 0 to positive infinity.