Just for the sake of discussion only...
What you plan to do with the rifle should be a determining factor in which brand you buy. If you want it for hunting or will only shoot it 100-200 times a year at a CAS event and at distances under 200 yards, then the more economically priced Armi-Sport or other low end brand of rifle will suffice without the additional expense of a quality tang and front globe sight. They are available through many dealers priced around $700 for the basic rifle. Some posters have commented on various BPCR forums about firing pin problems and soft metal in the receiver area in the past year on some low end rifles and that repair work on them is questionable. I have no experience one way or the other, only commenting on what has been discussed on other forums.
Again, the lower price is enticing if you will only shoot it every so often or simply want one just cuz you've always wanted a buffalo gun, and that is great. Now, that said, if you plan on participating in Big Bore/Silhouette/Mid & Long Range events where you'll put 1,000-3,000 rounds a year through the rifle (working up a load on the range, practice time and actual competition shooting), then a more durable brand such as Pedersoli is what you want since the level of quality and manufacturer support is far superior than the other lower-priced brands. Pedersoli rifles are solid, shoot great and maintain their resale value. They have won many contests around the world and continue to be in demand as a fine rifle. A basic Pedersoli 45-70 complete with Pedersoli Vernier/Soule tang and front globe sights will start around $900 m/l depending on the store. Check out the Pedersoli website (type in 'Davide Pedersoli' on your search engine) to see what is available and you'll get an idea of the high-end and low-end pricing and add-ons. Check the 'For Sale' or Traders forums and you'll see various Pedersoli's listed, that will give you an idea of how well they maintain their resale. You'll notice that the other brands do not.
The only other more popular rifle, but with a higher price tag, would be a Shiloh Sharps or a handmade rifle from the guys in the Southwestern US, the ones that are seen at Raton when the Big Boys get out the toys and go for the gold.
So, yes you can fill your needs with an Armi-Sport but a Pedersoli may fill them better over the long haul. Take a serious look at what your needs are and go accordingly, if you decide to go the route of competition then a high quality globe and Vernier/Soule tang sight are a necessity. Don't scrimp on sights, even a cheap rifle can shoot well with good sights and a better rifle will always shoot better with great sights. You might even find a used Pedersoli rifle complete with everything you want on a 'sale' forum at a price below that of new or near-new and in your price range.
Regarding your comment on the Billy Dixon and Quigley models, any Pedersoli rifle will do the job, these two are popular simply because of Old West history and Hollywood. Billy Dixon used a 'Big 50' they say and not a 45-70. The Quigley rifle used an "...experimental load..." as Marston said in the movie and was not a 45-70 as you seek. None the less, the design and styling of both are attractive but add a higher level of initial cost. I have a Pedersoli Billy Dixon model in 45-70 with the high-end Pedersoli sights, it is a beautiful rifle and shoots well. My Pard has a Cabela's-stamped Pedersoli with Ukranian sights from 'Parts Unknown' and it shoots better due to the sights and not the gun, they are identical except for the fancier wood and pewter nosecap. So, unless you absolutely have to have a Billy Dixon or Quigley, the lower end Pedersoli will shoot just as well as long as it has better quality sights.
I'm sure you'll get lots of other comments and suggestions here.
Best regards and good shooting!
'Ol Gabe