Bergen is very popular on BBO but in f2f bridge I have never played it, so I found this hand interesting. We had agreed reverse Bergen.
I have just googled and looked in BBO advanced and not found what the continuations to a Bergen raise are. So my questions:
When is a new suit bid by opener a splinter?
How do you bid a feature if you cannot splinter?
Are rebids the same after the intervening double as here?
If I knew for certain that partner's 4♣ was a splinter I would have bid 4♦ to show a control.
fwiiw without Bergen I see an auction that goes like this?
1♠ 2NT (raise to 3 or better)
4♣ 4♦
4♥ then key card check if you are feeling lucky.
4♣ is splinter with some slam interest.
Any suggestions for bidding the slightly better than 50% slam?

Help

Bidding:
p 1s p 3c
x 4c p 4s