EDIT: Thank you for all the links you provided answering the first part of the post, regarding the stacking of Polearm Master (PAM), War Caster (WC) work and Spell Sniper (SS) feats. [if someone didn't know, the consensus is that combining the three will allow you to cast booming blade in place of an opportunity attack when an enemy enters or exits your reach of 10 ft]
I would be grateful if someone could help with the second part regarding the damage output.
From a damage point of view, using the cantrips as the main source of damage, I will not allow me to use the PAM bonus attack. That is until level 7 when I use the cantrips until Level 7 where EK gets War Magic
Beginning at 7th level, when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.
However, I will also be losing the Extra Attack feature since I will not be able to attack twice when using a cantrip. Then scaling past level 10, I will lose even more Extra Attacks.
My questions are: a) Is that how Extra Attack and War Magic interact? In that regard, is it worth becoming an Eldritch Knight?
b) Other than thematically, does it make sense in terms of damage to use 3 feats for what appears to be a less than synergistic character, between normal attacks, cantrip attacks and extra attacks?
c) Should the natural progression be level 1 PAM, level 4 SS and then level 6 WC?
Again, thank you for taking the time to read this and reply!