Only way I was able to safely deal with Veterans was to already have good shield skills, such that I could slowly add bleed without multiple hits stacking up to the point where his passive would overload them.
Having a good shield base is pretty much a prerequisite to any good run, but I don't think the Veteran needs it especially.
His increase in damage only seems to go up per individual attack against his health, so speccing for power and doing a single strike against him every round doesn't boost his damage too much, and you can get in more hits if he's not doing a double attack. This does mean he can boost himself like crazy though if you get too many auto-damage things - thorns, the heart of pain, etc.
Bleed-focused builds are also especially effective because bleed damage only hits his HP as one attack at the end of the round, causing only one boost. I'm not sure if I'm getting all the card names right, but Injury (2dmg + 4bleed) can build the bleed counter early if you attack him with it when it won't pen his (small) shields, nor will Curse (2weaken + 4bleed), ever. Scuvy (+2bleed every turn) is just amazing in general. I also once got a card which doubled an enemy's bleed counter which is crazy good once you've got a few points on them.
That said... yeah, good shields are pretty much a necessity because there's few common ways to avoid damage or rebuild health. Seduction helps with groups by taking someone out for a few turns (and helps with singles by giving you some free turns to deploy the boost cards) but you still have to deal with the other enemies, and can't use it against bosses. The do-lower-damage cards seem to all be extinct and still let enemies chip away at your health if you don't have shields. There's no dodge chance to modify. You're railroaded into developing shield skills a bit, regardless.
