I have a couple of thoughts on how the shows could use this crossover.
If they don't want to actually intro Supergirl onto the ARROW or FLASH, why couldn't Melissa appear as the character of Karen Starr / Power Girl? It would give those shows a Supergirl type character without having to deal with why there is no Superman character. And Melissa may enjoy playing more aggressive character of Power Girl / Karen.
And over on SUPERGIRL, Kara could go to Central City on her Earth and maybe meet a non-powered Barry Allen. Or she could need to arrange a meeting for Cat with the mayor of Star City - Oliver Queen. And maybe this Oliver's mother hasn't been killed? And he is decidedly not Green Arrow. And maybe tries to get Kara into bed?? It would be a way fun episode!
Just a few thoughts.