Shadow Dragon and the Blade of Light
The Blazing Blade/Sword
Awakening
Heroes
First slot should go to one of the earlier entries. Shadow Dragon's the safe bet since not only is it the actual first, but it was localized (first the remake and later the original) while Mystery and Genealogy weren't.
Second slot should go to FE7 due to being the first localized FE. It was also very popular back in the day. Granted nowadays there's enough of a backlash against the game to put its slot a bit in jeopardy. But I still think it's important to have one of FE7-10 here and I'm not sold on any of the other 3 taking this slot.
3rd slot is pretty much a lock for Awakening. Like it or not (I don't), the whole "Awakening saved the franchise" narrative along with Smash Bros giving it Ocarina of Time levels of respect make it a safe inclusion.
4th slot is more in the air. Three Houses is probably the safe pick for being Awakening's bigger and better little sibling. But I don't think it had quite the same impact as Awakening did, so I'll be controversial and throw a nod to the never-ending gacha that's gotten IS a ton of money.
"Foolishness, Dante. Foolishness... Might controls everything. And without strength, you cannot protect anything. Let alone yourself."
- Vergil, DMC3