Nope, no class changing or customization.
There will be a limited amount of upgrades later, though. Some story/character-bonding paths will upgrade Felix (like giving him more damage, conditional counterattacks, or followup attacks from any angle with certain characters). There'll also be a shop where you can buy an expensive permanent upgrade, one per character, which is primarily intended for routes where you don't recruit everybody (and thus need your smaller team to be stronger).