I got my Fenrir sometime after I got all the minions and the Adamantoise mount from just doing King Elmer and GATEs, but with all the stuff to do nowadays, it's hard to justify grinding.

If you REALLY want it though, it's about the same amount of time I think for both King Elmer and Indolent Imperial but there's less grinding for cards with the Elmer strategy which is what I wanted to avoid. The drop rate for Gaius is so much lower than the one for Bahamut so I think it evens out in the end even though the MGP reward is slightly lower from Elmer. I also got quite a bit of MGP from doing Chocobo Racing, but my main goal was the chocobo helm and I kinda just ended up with 300k+ MGP on the way there.