From a perspective of a working man, I would say, price is the main issue here.
So, bolt action, is economic option.
However, reading many of the opinions and feebacks on this great forum, I came to conclusion that general concensus on this forum is, bolt action is for client, double rifle is more a back up rifle for PH.
Its not the rule, but the majority of opinions that I have read, tend in this direction.
However, if you can get good, accurate, reliable Double rifle, go for it.
Bolt action, in 375 will give you also an option for longer accurate shot.
In double, will more depend on how it is regulated, etc.... but generally, double is close range weapon.
And it will depend on what you will be hunting, and where. I would say, a bit more range you will get with bolt action.