I can't seem to find any good sources online for this, all I get are documents on how Nice model was used to compute Neptune's gravity capture of Triton via a binary dissociation, possibly because of similar terminology. So to the question;
To what extent, in terms of % delta-v, could Triton's retrograde orbit around Neptune gravitationally assist in slowing down a direct Hohmann transfer from Earth to Neptune for Neptune Orbit Insertion (NOI) during a single close-shave flyby of Triton?
I realize that the transfer would take about good three decades (I calculated 30.61652 years at Hohmann semi-major axis of 15.53545 AU using Kepler's third law and semi-major axes of Earth and Neptune), transfer time is not a concern, but I'm stuck with calculating velocity relative to Neptune at Triton's altitude for this transfer (I'm assuming Triton to be at the right place at the right time) and I didn't yet get to calculate delta-v that a low altitude (100 km above surface) gravity assist flyby of Triton could shed, so I'm not sure I'd know how to do that either. Neptune NOI would be for a Neptune orbit with semi-major axis of roughly 5,000 km above its surface (at 1 atm pressure). There wouldn't be any plane change for the target orbit.
This is not a homework question. It's been about 20 years since I last had to do such calculations back in the college years and I'd appreciate a bit of help in brushing up on this. I could plug this in some software but, call me masochistic, I wanted to do it the old-fashioned way. Back-of-an-envelope calculations would do, preferably discussing any shortcuts there might be in getting first-order approximations faster. No need to be too academic either, but I would like to see some calculations here, if that's not terribly inconvenient, in which case I guess a good reference with a short-ish writeup would do, too.
If you'll opt for the latter option using external references, please also include optimal launch dates, delta-v and Hohmann transfer times.