Positioning is key, even more so for this particular Caitlyn build. The general idea of prioritizing your headshot/R/Q dmg over spamming AA is to punish enemies disregarding your traps' amplifier for your headshots and having way more safe distance from key threats of yours in team fights. The extended headshot range from traps/net and your Q range are your prefered playing field to safely support your teammates. Use AA only if you don't run into the risk of getting bursted/cc'ed down obviously. Also, having the ability to execute back line champions around 1.300 HP with your R alone helps out hugely. I see killing the AP/AD Carry by positioning correctly and using my Ultimate as more important than AA'ing/Q'ing the tanky front laners if the opportunity presents itself.

Caitlyn's Q has wonderful damage for the early trades in lane. Getting the first-hit dmg of your Q on the enemy goes a long way and leads to having the priority early on.
Ranged harass combined with excellent wave clear enables you to either play aggressively or take the time to be useful for other lanes via ganking/securing objectives. With the tools in your disposal (traps/net) you should be relatively safe from ganks to the point of even punishing the enemy jungler hard for ganking you by net-headshot -> Q -> trap-headshot them to death/critticaly low health.

For any lethality build people usually run Duskblade as their prefered Mythic or Eclipse, because Prowler's seems kinda troll. For the most part you don't use the active and setting your focus on using it gets you killed. So why running with Prowler's? First the benefits of running the 'meta' off-meta picks for this build:

Duskblade:
+ Stealth on takedown
+ Passive Ability Haste

Duskblade's stealth seems useful first, but in actual gameplay I rarely find use for it if I position right. The extra Ability Haste on top of the usual build is too excessive and I rather have a more direct boost to my damage output.

Eclipse:
+ Omnivamp
+ Shield on engaging fights
+ Max-HP dmg
+ Passive %Armor-Pen

Eclipse's Mythic passive is great for a tanky enemy team composition. However, my approach to 'Lethality Caitlyn' is to get max value out of headshots with traps and your R onto squishy targets. Going for %Armor-Pen would mean I have to run more lethality focused items to achieve my goal. But by doing that I miss out on either extra on-hit DMG on trapped enemies, being able to utilize Infinity Edge to rise CritDmg or lower AD stats. Going for Prowler's gives you a huge lethality stat stick that provides 38 Lethality alone down the line combined with LDR is more than enough to threaten squishies and tanks alike.

Max-HP damage goes the same way, good agains tanks, but not my priority. If you are mid, you have an ADC/Bruiser to occupy them for you. You just need good enough %pen to be of some use against them or champs with above 110 Armor.

Shield on engaging and Omnivamp is nice to have, but even on low HP and without it you can be of great help. Your Q/R don't care if you are full HP or down to 1 to deal damage. It is more of a help in laning/solo'ing and solo'ing as technically an ADC is not the greatest of choices except you tactically prepared your traps for that encounter.

If playing safely and your trap settings are godly, Prowler's provides:
+ 38 Lethality on its own
+ 20 AH compared to Eclipse
+ Unlocked safe build pathing to IE's 35% CritDamage Bonus

Even the active of Prowler's has its unique (but rare) use cases that turn a losing to a winning fight.
For example:
You got transported to the shadow realm by Mordekaiser? -> dodge his Q bonks with Prowler's active/your net/flash and show him who the true king of this realm is

Qiyana closes in to you to stun and pin you onto the wall? -> place trap, use Prowler's active to dodge getting pinned to the wall and retaliate

Lee Sin getting behind you and wants to kick you? -> use Prowler's and let Lee Sin kick you to safety free of charge

Irelia jumping onto you and wants to stun you? -> use Prowler's to show her your dance moves and to dodge the stun, denying her Q reset

Sion charging his Q while you are slowed/no flash/no net? -> Prowler's behind him to avoid the long CC and run for your life

... and more niche use cases

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I could be wrong about my choice of going for Prowler's but after all my games in normal and ranked games in the range of Gold-Diamond I found my liking to Prowler's. Even if that means getting sh*t on by your teammates and causing an epidemic of eye cancer for players in my match <:D.