Nobody is going to let you enter their country with that. Are you also going to forge the clinic and lot numbers?
I'm talking about large events. I've seen it in person at a large regional volleyball tournament. The people processing hundreds/thousands are either (1)not trained (2)don't care (3)want to get people through their line and so forth.
As for copying lot #'s...big deal.
Sure, why not? Plenty of people have posted their vaccination cards online with that information showing.
Put the CDC symbol on the fake card and call it a day. Who's gonna check? How will they check? They won't, there aren't enough people to do it and there's no system setup for it.