Notes from author
Very similar to my Havel's Buff Quality Flip that can also be found on this site, the description of that build may paint a picture of how this is used.
The LC has amazing rolling attacks, and the left-handed axe actually provides a great rolling attack in the form of a regular r1 as well, whereas the right handed axe is good for defensive play.
When dual-wielding, a 20 poise strike from the hand-axe can allow for a stun on the amazing 50 poise 1hr2 so you can't be punished as long as you hit or are less likely to get punished against a toggler if used at proper reach, this 1hr2 can be followed up with a whopping 70 poise draining 1hr2 that has shorter reach but works well for catching bs attempts if done carefully. The 1h rolling attack or a simple r1 also drains 50 poise, so the hand-axe works well with those as well.
When two-handed without the buff, the LC is a bit risky. The only attack that seems fast enough for good poking seems to be the rolling attack, so you'll probably need to do loads of false flips so they come out unpredictably (Can't be parried, at least). It can get a bit boring, but you can actually use more of the move-set mostly reactionarily if your opponent doesn't have great pokes themselves.
While the buff is active, this build can use the LC's full move-set incredibly aggressively with much less risk, watch out for stamina though, no grass in this phase.
The rh hand-axe is just there for defensive play, it falls a bit short compared to dex weapons and won't win against a katana-flipper and certainly not against a spear-flipper (Spears counter the club too, like they do with most things) but it's a good finisher even against them.
Also, you've obviously got amazing roll-bs, I don't like to deliberately use it but it's certainly a good thing to have as an option in case of an opponent that uses similar tactics. Be wary that the very sight of a large/great club-flipper may actually cause them to do such things retroactively.