Bright guitars are much better than dull ones, so be glad you have a 'good' one ;)
You are using the tone pot on your guitar, aren't you? Backing it off to 6ish may be all you need to do...
Pretty much +1 to this from me.
I know it's all down to how one wants to use one's guitar (tone controls or not, etc), but this is exactly how I learnt how to do it when I was gigging.
I don't often say it in all the threads I join in where "too bright" is the issue, but my experience tells me that (for me personally) there's no real problem with too bright - just wind the thing back a bit on the guitar controls. The problem is when the thing ain't got the brights you need - you can fiddle with your amp and effects all you like, but invariably you won't get where you want to be. (And this applies for home use as well, not just gigging volumes, with drummers and wotnot, where it all became a lot more obvious and intuitive for me)
Oh, and by the way, even with the tone down to 6 or so, I always found some point during the gig where rolling it back up to 10 was still not bright enough! :lol:
Probably doesn't help you a whole lot at the moment.... It is something worth considering, though, when trying to figure out what pickups you want to get to where you want to be.
On the pickups themselves, I'm not that much help, I haven't tried Black Dogs, but that was my first thought from what I know of them - beef up the middle a bit. I was quite surprised to see a Mississippi Queen recommendation for this though - you'll definitely want to be using the volume and tones on those in a bright guitar! :lol: (And a very nice effect it would have too - I'm just not convinced it's what you're after right now!)