The counteroffer can easily be conditioned on seeing a copy of your offer. They don't need to see it today, but they might when you start your job...
But say you don’t want to forge one for just any company because that might not be legal. You could make your own company with a site and make offer letters that way.
FWIW I've done this several times (multiple offer scenarios where I have companies compete against each other for me) and never once has a company asked to see actual proof of the other offers on condition of hiring me. If that was a condition, I would say, sorry, I'll go with one of the other ones