Yes, rounded wings have been tested in the past. Probably the most famous example is the Vought XF5U "Flying Flapjack". They work perfectly fine. However, the design as pictured does have some flaws.
The lower the aspect ratio (i.e. wider front-to-back in comparison to their length left-to-right), the more inefficient the wing. This is due to there being a lot of area available for the high pressure under the wing to wrap around to the low pressure on top, creating large wingtip vortices. For the Batwing, this would be doubly true, as a large part of the wings have edges on the inside, almost doubling the effect.
It appears that the engine is either directly behind or directly below the cockpit, putting the center of gravity way too far back compared to the center of lift. This creates an unstable situation, especially at low speed. As the elevator loses effectiveness, the nose of the aircraft will tend to rise, further exacerbating the loss of speed and leading to a stall. Such a stall would be almost guaranteed to be unrecoverable using normal means.
(I say "normal means" because, in the show, the Batwing was shown to have VTOL capability, meaning it has vents in the bottom to redirect thrust down and hover. So, it would be theoretically possible to use engine thrust to rotate the plane nose-down and recover from the stall. If the engine fails, though, this airplane becomes an absolute deathtrap.)
Control Surface Effectiveness
Unfortunately, the animators of the show never animated the control surfaces, so we're left to guess where they are and how big they are. If they're in the standard position in back, that puts them close to the axis of rotation (which goes through the center of gravity, see above), limiting their effectiveness.
The elevators could be moved to the leading edge, but the slipstream on them would cause issues with flutter, and it would take a huge amount of control force to keep them from being pinned at their limits. This also wouldn't help the ailerons (because they'd still be close to their axis of rotation) or the rudder (because there's nowhere to put it other than that big mohawk structure coming off the top of the cockpit).
So, in short, it's possible, but not in any way practical.