Zen Archer, Flurry of Blows, and Empty Quiver Flexibility

[Hikuen]

Zen Archer monks "can make a flurry of blows as a full-attack action, but only when using a bow (even though it is a ranged weapon). He may not make a flurry of blows with his unarmed attacks or any other weapons."

The Empty Quiver Flexibility feat states "While using Empty Quiver Style, you can apply any feats and class abilities you possess that modify your ranged attack rolls and damage rolls with the chosen weapon to melee attack rolls and damage rolls made with that weapon."

Would this feat allow a Zen Archer to flurry of blows in melee using the bow as a melee weapon?
My logic behind this question is as follows:
1: The Zen Archer ability states that you can only flurry of blows with a Bow, which is what we are TECHNICALLY doing
2: The Zen Archer flurry of blows does not state that the flurry with the bow has to be ranged attacks
3: Flurry of Blows is a class ability, thus qualifying for the portion of Empty Quiver Flexibility that mentions feats and class abilities
4: In a follow up to #3, Empty Quiver Flexibility requires Stabbing Shot, which requires Rapid Shot. Rapid shot is a feat that adds additional attacks (same exact mechanic as flurry of blows, prior to ki points), and Stabbing Shot specifically mentions that the melee attack replaces the extra attack from Rapid Shot, so the precedent for flurry of blows to count as a "class ability that modifies your attack rolls and damage rolls" as stated in Empty Quiver Flexibility is already there.

So, what do you all think? Would this work? Can a Zen Archer be made into a flurrying switch hitter via these feats?

EDIT: It has come to my attention that the first feat in the chain (Empty Quiver Style) states that the melee attacks are "as if you are attacking with a Heavy Mace", which monks are not proficient in. Will this be a problem (- to attack/dmg?) or is that description only for purposes of knowing how much damage it deals, allowing the melee attacks to count as proficient based on bow use (seeing as requiring proficiency in multiple weapon types seems odd considering the point of the Style)

[RAWmonger]

Yes and no, it depends on how you want to read the text to Empty Quiver Flex. It says "While using Empty Quiver Style, you can apply any feats and class abilities you possess that modify your ranged attack rolls and damage rolls..."

Modifying rolls is different from modifying attacks in general. This feat is saying feats/abilities that give you a bonus (or a penalty, as i'll get to later) to your attack or damage roll...

However, flurry of blows does modify your attack rolls, it gives you a -2 to hit when you use it. And there's no text that says that you only use the part of the feat that modifies the roll, so technically you can use it with this feat, although it's my opinion that it's not meant to work with it.

To answer your edit, you only use the stats for the weapon (like when you'd use spear dancing style and the butt of your spear gets the stats of a light mace, you still get the bonuses from any weapon-specific feat/abilities of the original weapon).