Their are a lot of factors that go into deciding a conture , like how many rounds you plan to fire in a row becasue a lighter barrel will heatup faster and start to shot point of impact changes
How heavy of a gun can you stand , if this gun is going to be carried for a long way then you may want to choose a lighter barrel to help with packing it.
Is the gun going to be shot off hand , if so you want a barrel thats going to help ballance out the gun so its not to muzzel heavy.
Generaly I like a heavy barrel even for shooting off hand but I'm a little stonger than the average guy so it doesen't bother me that much but no matter how strong you are a realy muzzel heavy gun will not "swing" well..
Whats the gun going to be used for , for the most part a Rem Varmint conture will hold pretty good groups with non magnum cals for 5-10 rounds before it start to get hot and move.
For a deer rifle that generaly doesn't see more than 3 shots in a row I'd say a #3 or #4 conture would be good.
A good barrel maker can make a barrel that will shoot very accuratly with a very thin barrel , I had a 300 Wby with a #1 conture that shot awsome 3 shot groups if you could space the shots a min or so apart.