Now that's a lot of feats. But, the fighter does get a number of extra ASIs to use. And even without the interactions Polearm Master and War Caster are just good feats to have on an Eldritch Knight so the build performs pretty well before it goes online. I'm aware that it's probably not optimal dps, but if I cared for such things I could just go Dex with Sharpshooter. I find this build appealing because I've always liked gishes and halberds and mechanically I find this interaction pretty neat. I haven't seen discussion of this on these forums so what do you think about it?
I've done something similar (using Polearm Master, Warcaster, and Booming Blade) - but in my case it was a Paladin (Vengeance Oath)/Warlock (Tome) multiclass who used a Quarterstaff (no reach, so no need for Spell Sniper).
Tome Pact allowed me to get the Shillelagh cantrip as a Warlock spell and use Charisma for attack/damage bonuses with the Quarterstaff.
Oath of Vengeance - the level 7 ability allows you to move up to half your speed (without provoking an opportunity attack) when you hit a creature with an opportunity attack.
Polearm Master means they provoke when they come adjacent to you.
Warcaster means you can cast Booming Blade as an opportunity attack.
The idea is that they close with you, you hit them with Booming Blade as a reaction, and then immediately move away - forcing them to take the extra Thunder damage if they want to follow you.
It takes a long time to get all the pieces working together, but I found it fun.