On your names…
[name_m]Frederick[/name_m] → a handsome and charming name but [name_m]Freddy[/name_m] / [name_f]Fred[/name_f] do feel like the go to nicknames. Maybe less so within the family / your close circle who already know [name_m]Teddy[/name_m], but beyond that, he may be called that. To be fair though, my cousins have rhyming nicknames and it hasn’t been a huge issue… so maybe if he was [name_f]Fred[/name_f] – [name_m]Teddy[/name_m] and [name_f]Fred[/name_f] don’t sound too close/too much to me 
alternatively, I think you’d need another go to nickname in mind?
[name_m]Julian[/name_m] / [name_m]Julius[/name_m] → [name_m]Julian[/name_m] feels solid but flowing and gentle at the same time – it’s got a a touch of flair and fits with your set.
I do think [name_m]Julius[/name_m] does have a bit more edge – not over the top, just a tiny bit more presence? I still like [name_m]Julian[/name_m] though
[name_m]Alexander[/name_m] → classic and grounded, though I admit it doesn’t sparkle for me
[name_m]Benedict[/name_m] → bold but familiar. How do you feel about [name_m]Beck[/name_m] or [name_m]Ned[/name_m] as nicknames? Or even [name_f]Benny[/name_f] or [name_m]Benno[/name_m]? He may well end up being called [name_m]Ben[/name_m] but maybe less so if he’s got another solid nickname?
[name_m]Augustine[/name_m] / [name_m]August[/name_m] → I don’t think it would be bad to use [name_m]Augustine[/name_m] if you love it! though I do wonder if it would be it feel less ‘special’ now? Or would [name_m]Teddy[/name_m] like the idea of sharing something with his brother which could make it feel more special? Then all the bous would have a Au- name?
[name_m]Constantine[/name_m] → bold but familiar. I do think it’s rarer than your kids, but it doesn’t see at odds with their full names. What would be your nickname choice?
Some more ideas…
[name_m]Nathaniel[/name_m] (Nate, [name_m]Thaniel[/name_m], [name_m]Nat[/name_m])
[name_m]Alasdair[/name_m] or [name_m]Alistair[/name_m] (Asa, [name_f]Aster[/name_f], [name_f]Sasha[/name_f])
[name_m]Dominic[/name_m] (Nico, [name_f]Donny[/name_f], [name_m]Nicky[/name_m], [name_m]Doc[/name_m])
[name_m]Valentine[/name_m] (Lennie, [name_f]Val[/name_f], Vinnnie, [name_f]Len[/name_f])
[name_m]Solomon[/name_m] (Sol, [name_m]Solly[/name_m], [name_m]Sonny[/name_m])
[name_m]Cassius[/name_m] (Cas)
[name_m]Raphael[/name_m] (Raffy, Raph, [name_m]Rafe[/name_m])
[name_m]Ambrose[/name_m] (Bram, [name_m]Brosie[/name_m], Ro)