Producing a print layout using 2nd map view in QGIS

After some more experimentation, it can be done but the trick is to not use either of the polar projections for the map canvas. Instead leave the main map in epsg:4326 and then in the two maps in the print composer set EPSG:3413 and EPSG:3986 and all will be well.

I think this can be done without the second data-frame. If i understand your question correctly, i would just add as many map frames to the print composer as you need, pan and zoom them in place and lock them when they are ok. All with just one set of themes in one map view...