The focus of the combo is opportunity attacks, though. Stacking booming blade with Spell Sniper, War Caster, and Polearm Master means you can make an OA with booming blade any time an enemy comes within 10 feet of you. So you're already doing bonus damage on the OA, and then you add more bonus damage if the enemy closes to melee. (And if they came within 10 feet, they probably wanted to be in melee, so you're either dealing the extra damage or denying them their preferred strategy.)What you are wanting to do will work, but it is inferior to the normal option.
Then on your own turn, you attack like normal.
It's a neat little interaction. Polearm Masters make a lot of OAs, so if you have the feats to burn, this makes them significantly better. And the best part is that you don't need all three pieces to get value out of the combo. Polearm Master, of course, is an excellent feat on its own. War Caster gives you advantage on saves to maintain your concentration buffs, and you can take advantage of Polearm Master OAs to cast offensive spells without wasting a whole action. So you aren't sacrificing effectiveness at lower levels before the full combo comes online.
If it were me, I'd go variant human (if allowed) with Polearm Master; pump Strength until level 6; War Caster at level 8; Spell Sniper at level 12. If you were really determined, you could get the combo as early as level 6, but then you're sacrificing Strength boosts, which is probably not worth it.