Rules-as-written, looks like yes.
I'm not sure that it matters that the character is an Echo Knight; the same question would arise for any player with a two-handed ranged weapon and the Thrown Weapon fighting style.
Relevant rules excerpts and references:
Two-handed weapons: PHB, p147
This weapon requires two hands when you attack with it (emphasis mine)
Thrown Weapon Fighting: TCoE, p42
You can draw a weapon that has the thrown property as part of the attack you make with the weapon [no mention that the attack has to be ranged]
Other Activity on Your Turn: PHB, p190. Drawing or sheathing a weapon is explicitly mentioned as something you can do in addition to your action (which is relevant for putting the weapon away before attacking with the bow again).
Opportunity Attacks: PHB, p195:
you use your reaction to make one melee attack against the provoking creature.
It's not specified anywhere that you need to currently be wielding a melee weapon in order to make that attack, just that the attack itself is not ranged.
So, you don't need to hold a bow with two hands when not attacking with it (i.e. outside of your turn), you can draw a weapon with the thrown property as part of an attack whether you're throwing it or not, and you can sheathe the weapon again on your turn before attacking with a bow.