If I am not mistaken, the MB5 uses the turn signal idiot indicator to ground the active turn signal through the non-active one.
i.e. you are backfeeding each blinker through the idiot light. and i think it is designed to trip the thermal breaker in the flasher unit before the other blinkers have a chance to light up. could be wrong on that, but regardless, there are two things going on you need to fix, and probably a few ways to skin the cat.
as people have already stated, get a flasher unit rated for LED draw. the LEDs don't pull enough current to trip a standard flasher unit. resistors or ballast units could also get your system working without any creative re-wiring.
What i would do is get a 3-prong LED flasher unit and modify (or delete) the idiot light wiring. these have a third conductor that sends juice to the idiot light on an isolated circuit. because your idiot light is wired in to both hot sides of the blinkers, you would need to chop it out of that circuit and feed it with the 3rd prong signal, and ground the other side. if you don't need the idiot light, just run a 2 prong LED flasher, but you still have to disconnect the idiot light or everything will light up.
the other thing I have done in an existing wiring scheme using a 2-prong, is use diodes to feed the idiot light. and again, depending on the idiot light wiring, joining the two aux legs from the blinker circuits AFTER the diodes, and making sure it goes to ground on the other side of the bulb.