This should perhaps be a comment rather than an answer, but I thought I should post it so that others can offer corrections or further detail.

One of my colleagues has sometimes justified his opinions/suggestions to me with references to the Scimago journal rankings in mathematics:

https://www.scimagojr.com/journalrank.php?area=2600&type=j

Unfortunately I don't know if these rankings carry the appropriate "weight" in the eyes of deans and similar creatures. (I also don't know about accuracy, but your post suggests that this is not really the main issue which you want to address; the ranking puts JAMS at number 3 and Annals at number 4, for what it's worth.)

There's a lot of finer-grained detail one can extract from this database, by selecting a particular country or a particular area of mathematics.

Some information which may be of interest to readers other than the OP For background on this ranking or system of metrics, see https://www.scimagojr.com/aboutus.php

The SJR number is described in technical detail https://www.scimagojr.com/files/SJR2.pdf and it appears to be a metric based on a version of the Google PageRank algorithm: some "alt text" on the webpage claims tha the SJR number

... expresses the average number of weighted citations received in the selected year by the documents published in the journal in the last three years.