Lake Sump in Peak should be sumped now, it has airspace periodically but you can still treat it like a full free-dive in that case.
Go into Valley Entrance and do the first free-dive to Rowten (Usually 2m long, but again I've seen it low enough to be a duck)
Neither of those will need any lead, or even a mask.
Then I'd say you need to invest in a mask, some lead (1x 2kg block will probably do to get you neutral/negative, they thread onto a standard caving belt. Ask a cave diver if they have any spare, you might be in luck
), and possibly a neoprene hood.
Once you're comfortable in the water you're probably good to do the second sump from KMC to Rowten, and if you take that in your stride you can commit to the final 8m one - be warned there have been reports of debris in there after big floods. I've been in 3 or 4 times in the past couple of years and not seen any obstructions myself - though on my first attempt I did turn around after being spooked by a big trout swimming into me! It's handy to get a diver to check for you beforehand, or ask someone that's been through since the previous flood event. It isn't any more hard or technical than the previous one, but it does overstay its welcome so you just need to keep a level head and trust that the rope you're pulling along does eventually surface to air again!
Other good free-dives to tick are the Old Ing ones, which have a more technical profile - I think one is 2m long and 2m deep, and the second is 1m long but 3m deep underneath an archway, which is quite scary on a single breath! Obviously there's ongoing access issues at the Birkwith Caves, so keep that in mind.
Then of course there's the infamous Langstroth free-dives. They're very close to the bottom so you can try them out before doing the full through trip. Last August I helped re-bolt and re-line them with SRT rope, last month I did the through trip and the lines are still pristine. Be aware of the small airbell between the two most downstrem sumps, if you've not read any incident reports of the fatalities in there then it's probably best to familiarise yourself with the risks involved with the accumulation of CO2 in that airbell if it's been travelled a lot. Many people doing the trip treat the two sumps as a single dive and don't surface in the airbell, and some just take a single breath before moving on.
All of these won't require any specific free-diving apnea training, I'd say the main obstacle to overcome is just keeping calm and being comfortable with being underwater (and cold!).