The "try with the cover off" is a good suggestion since these are known for developing exhaust leaks. However, this sounds more like it COULD be that you have a dead or "low contribution" cylinder. These engines WILL run on one cylinder and WILL be smooth enough to fool you into thinking they are both firing. I would try a new set of plugs (if you haven't already) and check that both cylinders are getting spark. Checking these for spark is done easily if you have an induction timing light and a spare 12 volt battery to power it. If you're not set up for that, you can do it visually by removing and grounding the plug in the "old fashioned" way but the spark, even when its good, on these is very faint and hard to see in the daylight. Done in a darkened garage or with a blanket over your head and the outboard is a good way to go. Nighttime testing works too.
If you verify spark is good then it might be time to perform a compression test. You should verify that the cam is in time and that the timing belt is snug and hasn't "jumped" a tooth or two as this could cause the engine to be "weak" but still run. If compression is low on either or both cylinders you will want to check the valve adjustment. If the valves are properly adjusted but you still have low compression, then you likely have cylinder ring issues. It might not be a catastrophe though as it could be just some carbon buildup has caused the rings to "stick". Since the engine runs, you could try performing some "decarb" treatments to free them up before you think about going into the engine.
And of course it could just still be a fuel delivery problem. I wanted you to try pumping the bulb to rule out a fuel pump problem and you have. So, You didn't say how it ran if you tried running it choked. If that improves things then I would definitely suspect carb. But if it doesn't run better with the choke out then I would rule it out.
Good luck.