Hi-Res patch fixes the lighthouse bug as well

It's possible to do it without the Hi-res patch, I think, although it's hard. But, I was playing a disc version so I needed to use CPUgrabber to slow down my processor to get the game to work. It's possible that that changed the behavior with the lighthouse jump.

I've heard that crouching mid-jump can work too.

I found some advice on the Internet that did the trick without any CPU programs – when jumping toward the ledge, make sure the following three keys are pressed: the move forward button (up arrow or W), the jump button (left mouse button), and the crawl button (spacebar). Then Cutter should grab the ledge and pull himself up. This did the trick the first time I tried it. If it doesn't work, change the jumping distance from the edge or the angle a bit.