Hmm, there shouldn't be that much input lag for DMC1. Admittedly I've only played it on the PC version, but I seriously don't see them putting the game on PS4 with that amount of lag. It could be that the game seems to be slow to take inputs, it is an old game after all and it's definitely significantly slower than DMC3 for example. There's also combos you can only pull off by timing your button presses, which slows the pace of combat a bit.

If it is that bad though, you'd be fine to just skip to DMC3 which I am sure should not have any input lag issues, and it's just such a fantastic game that's worth buying the collection just for that one game.