@OP you can mullet it without an issue. It's just a wheel.
Only thing you'd need to retain is the same cassette (for chain length and gearing compatibility) Hub width, axle size and disc size.
You'll be dropping the BB by about 20mm or so, but that's not a train smash unless you're already grazing rocks with your cranks & chainring. It'll also reduce the inertia on the rear, as long as you're not going heavier with the back wheel. That will make the rear more "lively" by virtue of it being easier to change direction, but not by much. Another by-product of that is that you'll be able to lift the front more easily as it'll have slackened out the HA by dropping the rear axle relative to the front axle.
FWIW I'd concentrate on the skills before you decide to pay them bills. Maybe even a better bar (slight riser) or a shorter stem. Flat bars belong in the gym.