I like the suggestions of [name]Lila[/name], [name]Willa[/name], and [name]Eliza[/name] for you. I think they all work with the MN [name]Catherine[/name] (which I think I saw somewhere that you’d like to use it as a MN?), and I love the idea of [name]Jack[/name], [name]Claire[/name], and [name]Lila[/name]; [name]Jack[/name], [name]Claire[/name], and [name]Willa[/name]; or [name]Jack[/name], [name]Claire[/name], and [name]Eliza[/name] together.
My first idea was [name]Ella[/name] [name]Catherine[/name]–[name]Ella[/name] is fairly mainstream but I think [name]Jack[/name], [name]Claire[/name], and [name]Ella[/name] are perfect together. What about:
[name]Jack[/name], [name]Claire[/name], and [name]Sophie[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Lucy[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Poppy[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Mia[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Eve[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Gemma[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Julia[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Olive[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Ada[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Violet[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Adele[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Fiona[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Lily[/name]
[name]Jack[/name], [name]Claire[/name], and [name]Hannah[/name]
Good luck!