It's easy to get D2SE to work, but it's less convenient than usual.
Instead of extracting a whole Median folder to your MODS directory, exchange the median copy of patch_d2.mpq for that file in the 1.13c folder under CORES (you can find that folder at the same level as the MODS folder). Then when you run D2SE, select 1.13c and it will of course find Median's patch and run that instead.
It is a lousy workaround - the point of D2SE is to make switching between mods easy and this is going back to manual replacement of files. But the point is, you don't have to uninstall D2SE to play Median, and you can still use its launcher.
Be careful to back up the actual 1.13c patch_d2 file!