Nicola Sartori - Median bias reduction of maximum likelihood estimates
For the 1-interest parameter logistic regression question, when the interest parameter has a finite estimate, you can reduce the regression problem to a simpler one with the same conditional distribution for the sufficient statistic associated with the interest parameter that removes sthe parameter estimated at infinity. The conditional likelihood remains the same, and so you could use this to reduce median bias in a context that doesn't have to worry about the infinite estimate. A reference is Kolasa (1997) Scand J Stat. Have you considered this approach?