As I mentioned, the RS-1's can be a finicky beast too much warmth and the bass gets mushy, while the opposite can bring cause the treble to get too harsh. IMO, finding the right system for the RS-1's is about balance, they take well to tubes but not the stereotypical overly warm tube sound.
The first couple of setups I had with the RS-1's was a Cambridge Audio 640C or Rega Planet 2000 paired with a Singlepower PPX3. With the Cambridge the treble was a bit too aggressive and sterile for my tastes and the Rega setup just didn't quite cut it for me, in the long run I think the PPX3 was a bit too tubey. It did alright but I felt like it was never able to fully grab the RS-1 and control them.
I moved from that system to the all-in-one Eastern Electric MiniMax which is a truly wonderful CD player which has a tube output stage that also feeds a decent opamp based internal amp. This proved to be an interesting pairing because the RS-1's got a dose of tubes but not overdone. I was able to fine tune via tuberolling hitting on a brand that worked quite well. In the end though the internal headphone amp started to show it's limitations so I reassessed my system thinking I could build around a better amp. That led me to buy a HeadAmp GS-1 which paired with the MiniMax CD player did a phenomenal job with the RS-1's. Unfortunately I couldn't afford to keep both so I sold the MiniMax thinking I might be able to get similar performance out of something less expensive. This never panned out for me, the couple of DAC's I tried fared ok but nothing ever really did it for me like the GS-1 and MiniMax pairing. Having to split up that combo is my biggest regret because I think I could have happily stuck with the system for a long time.
I have briefly heard the RS-1's in a couple of other systems, one a custom modded tube amp and source that were built originally around the RS-1 and later fine tuned for PS-1's. The RS-1's sounded quite nice in this setup but paled in comparison the PS-1's.
Another amp that I heard briefly with the RS-1 was an EarMax Pro fed by a Lavry DA-10 DAC. For the short period of time I got to hear this setup I was really impressed. Not quite the control of the GS-1/MiniMax combo but it had a more fluid, sweet sound.
If you're system is computer based I've heard that HeadAmp is planning a DAC only version of the Pico, I'm not sure what pricing will be like but I'd expect it would be about the same as the amp only version. The Pico is really a wonderful DAC at an unbeatable price, easily the second best source I've owned and I've often wondered if it might even better the MiniMax and it's a fraction of the MiniMax's cost. That would allow you to invest a little more in the amp end of things, depending on you're preferences I suspect a GS-1 or EarMax Pro would work well.
Of course there are other amps that work well the RS-1's I just haven't heard any and it's been a couple of years now since I owned them myself so I'm not as up to date as I once was.
Edit: I just noticed you mention keeping the budget to under a grand, is that including the phones? If so you might have a tough time going the RS-1 route considering they'll eat up about half the budget. As I mentioned I'm not sure on the pricing of the DAC only version of the Pico but I expect it to be in the $350 price range. A new GS-1 is $750 I believe but can be found a bit cheaper on the used market. The last time I looked I believe used the EarMax Pro's were going for around $450-500 but then you might need to spend a bit to get decent tubes.
Edit #2: A cheaper amp option might be Grado's RA-1, this amp gets a lot of flak because of it's relatively cheap innards but the fact is many people, myself included, feel it has a nice synergy with the RS-1. Of course it's not going to be near the level of the others mentioned but I always liked it with the RS-1's so I think it would give you passable performance until such a time as you could afford to upgrade.
