Can linear and logistic regression coefficients be combined using an inverse variance weighted average?

Inverse-variance weighting makes sense for combining parameter estimates that are in the same units. The problem is that linear and logistic regression coefficients are in different types of units. Per unit change in the same predictor, a linear regression coefficient represents the associated change in a numeric outcome, while a logistic regression coefficient is the change in log-odds of a binary outcome. So any average of a logistic regression coefficient and a linear regression coefficient wouldn't make sense.

There might be some methods from meta-analysis that somehow combine information from such fundamentally different types of studies to provide some overall measure of "importance," but I'm not familiar with them and the combination wouldn't be an average coefficient in any event. Perhaps you could do a crude combination by converting linear-regression results to predictions for an equivalent logistic regression based on a cutoff in the linear regression response variable. But offhand there would seem to be multiple problems with that approach.

It looks like this difficulty in combining different studies on the same phenomenon is another argument against dichotomizing continuous variables.