I've recently discovered a refights skip in MM3 that allows you to skip 7 of the 8 refights. I'm not entirely sure how it works, but I did include steps on how the perform the glitch in the video's description.
https://www.twitch.tv/videos/542566981
The key takeaways I've found so far is that Rush Jet causes havoc with the teleporters. I've been able to store a teleporter's function by riding over them with Rush Jet and then if you slide, the teleporter will trigger. Another key aspect is that where you'll appear after the 25 second black screen seems to be based on the last teleporter you used. That is why I fight Magnet Man in the video. If you fight Needle Man instead, you'll actually appear closer to the final teleporter of the stage, but you'll be placed in a wall that you must clip out of in order to reach the final teleporter.
This was done on Fusion 3.64 using the Japanese rom on a save file that had Rockman 1 and Rockman 2 cleared.