No lock washer is needed. You need a nut with a crimped end so it doesn't vibrate off. Or a nylon-insert type of nut. The lock washer is probably digging into the wheel.

Can you take the nut off the opposite side and take it to the hardware store for a perfect match?

Yes, you need a self locking nut. Do not use a lock nut.
If you cannot find one just use some fingernail polish on the threads. It should keep it tight.

A 3/8 course thread(16tpi) nyloc nut with a flat washer should do. BTW ,if you can't roll the mower with the nut tightened securely, I would guess you have the wrong wheel.

Another way to stop a nut from backing off is to use a prick punch to booger up the threads sticking out just beyond the nut.

Take the wheel to a lawn mower repair shop and ask for the correct locking nut. At the outside it might cost a buck and you will get the right part. Home Depot can in rare instances be a good source of technical
advice but this is not one of those times.