Elizabeth is a good choice, so is Carmilla. Carmilla was my pick and I've gotten decent use out of her, she has Charge Drain and dmg bonuses vs female characters (and there's a lot of those). There seem to be more options for Assassins than Lancers down the line, but that might not necessarily matter due to rarity. You have to actually pull them after all.
After using Carmilla as my first starter up to lv. 50, my conclusion is that you don't actually need a powerful assassin yet.
Right now, assassins are support with low damage and high star generation. Carmilla is a weird outlier with very high attack and lower star generation, so she's not great at either. You can only get frontline damage out of her if you build the party around spamming her (powerful) NP.
Who should I get from the Recruitment event to round this out?
I'm thinking either Seig/D'Eon or Elizabeth.
Your party headliners are Gil and Herc, so you want someone to support either of their strengths.
The absolute best would be Lancelot to feed Gil stars and to buster chain with Herc, but he's not a free pick.
Siegfried is just irredeemably awful, being overspecialized with far too much HP and far too little attack. D'Eon can eventually act as a tank to draw attacks away from Herc, but you need three ascensions to get that skill.
I like Elizabeth as a versatile pick that fits into any party. Here she can pile on double attack buffs alongside Gil and debuff a boss, plus both her attack and HP are solid.