It's correct that "Sehr gerne." can be translated to each of those English expressions. While those expressions indeed have something in common they also have subtle but distinct meanings. So at the end of the day it clearly depends on the context.
In your example, the German expression "Sehr gerne." is simply a variation of "Ja, bitte." and, as such, is translated to "Yes, please.". You politely accept an offer that was made to you.
Similarly, when you accept a request of someone (like, accepting when someone wants to sit down next to you), you can go with "Yes, please" / "Sehr gerne.". In that context, however, "With pleasure" is an equally valid response and so is the German "Mit Vergnügen.". (Though, this isn't commonly used).
The expressions "You're welcome", in contrast, is used as a reply when someone thanked you for a favor you've just did. This is usually translated as "Gern geschehen.", sometimes shortened to "Gerne." or "Sehr gerne.".
So, here's a dialog with your example where "Gerne" is used in two distinct ways.
A: Soll ich dir eine Bahncard mit bestellen?
B: Sehr gerne. (=> Ja, bitte.)
(1 Woche später.)
A: Hier, deine neue Bahncard.
A: Sehr gerne. (=> Gern geschehen.)