I've never serviced a watch but want to try. The materials will cost more than the victim. This particular subject is the most confusing of the lot so have spent a lot of time searching for technical manuals on movements.
It seems a service should involve replacing the entire barrel assembly these days. Only exception and it may be old details is the 7750-7765 where they just suggest a smear of a rather expensive braking grease on the outside end of the spring. I assume that lubricates the barrel as the spring slips. 8217 is said to provide normal braking on any barrel. A lot cheaper than Kluber P125. Interestingly they don't use many lubricants. Just 9010, 941or9415 and D5 which can be replaced by HP1300 according to some.
ETA use HP-1300, 9504, 9010, 9415. Also coloured HP-1300 on the 2824. Entire barrel replaced. It would be interesting to find much older details.
Info on older Seiko do give details. 2 oils S2 and S3. Not on the spring itself but on the roof and edge of the barrel. S3 if there is a black ring. It seems this one was a grease with an additive. Sounds like an 8200 equivalent, S3 maybe with graphite. The moly one can have problems with that settling into the bottom of it's container so personally I would avoid it. They use S6 which is a silicone elsewhere in newer movements where details are given. Cousins stock it. I've only looked at the NH3x. Also S4 and 9010, 9010 even on the pallets and in the usual places. I can't see any use of S4 on this movement so probably used on another.
Lubricating the spring itself? Mark uses 8200. The www jungle varies a bit. Too thin, use one with an additive. Few drops of some oil on the spring. None and braking on part of the spring makes most sense to me or maybe 8200 etc on the rest as well.
While looking around I came across thisMoebiusLubes.pdf and found it useful. It mentions a number of makes. One intended for springs is based on paraffin wax. Pass but it takes a long time for candles to degrade. It's pretty stable stuff.
I have found NOS complete barrels on ebay but they come from the japanese market. Not always available but they do pop up from time to time. Some of the aftermarket springs dont come with a washer that inside diameter fits the barrel.