On the basis of the simulations that you have done, I think the easiest solution is just to consider a simple logistic regression for each of the three explanatory variables separately - you can then still compare the regression coefficients for these three (scaled and centered) explanatory variables
On the basis of the simulations that you have done, I think the easiest solution is just to consider a simple logistic regression for each of the three explanatory variables separately - you can then still compare the regression coefficients for these three (scaled and centered) explanatory variables