You have essentially three challenges: Removing Kakugo (and DNA Tracker), Removing Ben Musashi, and Removing IPO.
Let’s cover the simplest: Removing IPO. This means you need another answer for Money in the deck, and Restructure is good, but I’d definitely bring it to 3x. (Probably in favor of the Interns you added) One of Jinteki’s weaknesses for the longest time was economy. It’s why RP + Sundew was able to get off the ground: It fixed Jinteki’s economy issues in one easy card. (Celebrity Gift didn’t hurt, either.) Sweeps Week isn’t countersynergistic; You’re generally forcing the runner to hold cards in their hand, so Sweeps is at least a Beanstalk, usually a Hedge Fund. If we run out of influence elsewhere, we can look at replacing them with Beanstalks, though.
Next, Ben Musashi. This one’s a bit harder. The point of Ben was to force the Runner to go even slower than PE’s natural ID power does, along with some amount of faking. Prisec can work here, and Hokusai Grid as well (bring in the third one). If we find some influence, you can try Red Herrings as well.
Now, the ice. Kakugo’s the big standout here, because it enabled this sort of ‘slow squeeze’ PE to exist, where you can win simply by having Kakugo on your important servers and the Runner out of cards. It’s repeatable, reliable net damage. I’m not sure you can replace Kakugo, it’s much more than an End the Run ice. You’re correct that you don’t want to aim for program trashing, preferring net damage instead. The problem is that I don’t think you have a good way to make them lose a card on each Successful run without Kakugo. The most similar would be Data Raven, but that costs them money+time, not cards. You can try Data Mine, it’s almost guaranteed, but it isn’t repeatable. Definitely go up to the full 3x Neural, though. (One thing to watch for is that this could encourage holding them all to try and just kill the runner, which isn’t the point here, and if people start doing that a lot, you’ll want to take one of them out.)
DNA Tracker was the big punishment if the runner starts running without a Code Gate and a Sentry breaker. Lockdown doesn’t really replace that. Crick and Merlin are the only ice that are Code Gates and relatively punishing… Unless you have Fairchild 2.0 or 3.0? 2.0 should’ve been in second Flashpoint pack… Use that if you can, that’ll satisfy. (The point isn’t so much that you want to be Yog-proof, it’s that you want to hurt them if they run without a Decoder. DNA Tracker happened to do both, so it’s great, but we don’t have access to it.) Tollbooth and Archangel would also fit here, but their effect isn’t quite as strong.
For your changes, the only thing I really disagree with is the removal of Eli. Leave that card in, this is a Tax The Runner Till They Stop Moving sort of deck, and Eli’s phenomenal for that.
I agree that Marcus might not fit here (Original deck has 0 psi games, partially because I think TBB doesn’t care for them, and partially because they aren’t one of the Basic Strategies the set of decks was focused on improving.)