You have a couple of options...Here is what I would do.
I would send that receiver, stock and forend back to H&R and have them install a 20GA USH barrel. But, you would have to mount a scope as the USH does not come with sites.
If you want a 12GA, I would opt for a Tracker II with rifle sites. You cannot mount a 12GA USH heavy barrel on a SB1 shotgun frame. But, H&R just came out with a Ultra Light Slug Hunter...That 12GA will fit an SB1 shotgun frame. I do not see it listed on the accessory barrel page yet. You may have to call for that one.
As far as effective range, I have heard of guys taking shots at 125 yards with both a 12 and a 20 with success. I have not had an opportunity to shoot that far and the farthest I have shot while hunting targets is 100 yards...I wouldn't stretch myself much beyond that. But, that is me and not the gun as the limiting factor.