You'd only need lightning integration and paying a small amount of sats, recycling the Bitcoin proof of work instead of making more. You could even have the server transfer the same sats back and forth as a token as long as the server side is happy.
You will not get your visitors to buy Bitcoin and set up a lighnign wallet to visit your website.
But having some JS on your site that crunches numbers for 2 seconds before the user can progress would work.
You don't have to buy it, you could crunch numbers for an equivalent cost if that's preferable. The advantage is that the effort can be stored and used later, so you need not even add a 2 second latency. Similar to "Privacy Pass".