Lube--
"Buy one"-Ughh. Totally missing the point.
Follow the k.i.s.s. method on this one. Skip the clutch-too hard to find one that will adapt to your weed whacker. Use a three point pulley and belt drive system. Attach one pulley to the weed wacker, one to the axle, and one that will take up slack on the belt to engage/disengage the drive. That pulley is mounted to a lever that will lock in place once you get it "over" the highest tension of the belt and back down the arc of movement-if that makes any sense. Think of engaging the belt drive on a mower deck on a riding lawn mower-same thing. If you can weld, this can all happen pretty fast.
This will make you Cool Dad of the year in no time-Or crazy Dad depending who you ask
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Btw-skip the chassis and brake upgrades. That stuff is for *******s. Keep it super low tech-I would mount the motor to the cart with duct tape and use twine as the throttle. Hell-I'de even leave the chain and pedals hooked up to add some "thrill" factor to the ride
Okay-maybe not that.
Have fun!