Ok great. Thank you for the guidance!
I have previously defined the prior distribution for control group as a Beta distribution (shape parameters: ⍺ = 3, β = 14) with mode 0.2. The primary analysis is estimation of a proportion so I think beta priors are appropriate, but for a secondary analysis I want to compare the proportion between two groups.
I set normal prior for the coefficient pertaining to the grouping variable corresponding to the log-odds ratio of 0 and SD 1.175; this gives 95% central probability mass between odds ratio 0.1 and 10.
To set the prior for the intercept in the logistic model I took a normal approximation of a Beta (4,13) distribution. The normal has a mean log-odds -1.26 (which corresponds to an outcome probability of 0.22) and standard deviation 0.601.
Edit: and after going through this exercise, I found 6.10.3 in BBR which is outstanding.