Since you are indeed planning on using this for hunting, then
choosing Sendero/RemVar class barrel contour or similar as
mentioned above is good advice. A Sendero class bbl can
be carried/ hunted with, and will still hammer tight holes from
a bench just fine.
As far as barrel brands, I am sure plenty of shooters have
their favorite and would name one in particular if they had
to pick one, but there are many great barrel manufacturers
nowadays. Choosing the right competent gunsmith to do your build
and install your barrel to suit your needs is probably just
as / or more important than barrel brand, IMO.
Barrel availability is another factor when considering your build.
Do you want to wait months for a specific brand or type barrel?
Some guys do, I probably would not, because I am not partial
to just one brand barrel. If my build had other components
that create a wait time, such as an action or stock not completed
yet, then perhaps I would be just order a barrel as well.
But since you have everything in hand already, and the only
thing left to acquire for your build is the barrel, then I would
look to source the barrel from someone who has one in stock
that meets your criteria. And start there.
Barrel blanks often time pop up here for sale at reduced
prices, making them a bargain, because they available
now and priced down some.
Since you have everything but your blank, I would find
one in stock somewhere and not focus on brand name.
Please do advise when you get it built with pics!