Median stacking and its uncertainties

So, I want to compute a quantity $A(sample) = median[\{A_i\}]$ of a sample. I'm using the median to penalise outliers. Usually I bootstrap the sample to calculate the uncertainties on the median, but each object in the sample has its own uncertainty associated with it.

It seems a waste to disregard measurement errors but bootstrapping the errors along with the measurement seems incorrect.

I suppose I could assume a gaussian PDF for each measurement based on its uncertainty (which assumes gaussian distribution anyway since it's just a 1$\sigma$ error). Then the median of the product of the PDFs would give me an answer. But this also seems wrong.

Any ideas? Which is the correct way to go?

Thanks

(I've looked at this but my situation has an emphasis on the median of the sample)

(this reassures that bootstrapping is fine to use with medians but that's it)