I have just performed extensive testing and found the following (bye bye hundreds of runes, runicite and diamonds).
Yesterday I had issues beating a 195% movement speed with my 200% movement speed Karaoke King (same team of Dragon Slayer, Dungeon Man, Karaoke King, Moon Drake, Void Wyvern). I ended up swapping out 1 rune dropping his movement speed to 184% which did the trick.
Today I had an issue with a 172.8% movement speed Karaoke King. I tried both 200% and 184% with no luck. I took notice that my Dungeon Man was running out much quicker then his (I had about 80% movement speed on him) and thought it may be a targeting issue. I swapped out my Dungeon Man and then Dragon Slayer to have 0% movement speed, no luck.
I then swapped out another rune which gave both my Karaoke King and the defending Karaoke King 172.8% movement speed (Dungeon Man and Dragon Slayer were still at 0%) and with everything equal, his Karaoke King fired first.
I then made one final attempt swapping out another rune, lowering my Karaoke King's movement speed to 149%. Finally it worked.
That said, I really didn't locate the "sweet spot" but think I eliminated some ideas.
Movement speed of other characters does not appear to have any effect on when Karaoke King's white fires. In a team of the exact same makeup, the defender's Karaoke King appears to fire first (at 172.8% movement speed in this case).
Not sure what else to test at this point but interested if anyone else got insight out of these results.