What is your budget and what type of shooting do you want to do? Sit at a bench, free hand, prone? It all makes a difference.
We know it isn't Vudoo, but if Bergara is in the mix, it has a couple of options. They have the B14R which is the Rem 700 footprint which opens you up to options later for stocks, triggers, etc. It however uses AICS type mags which can be a bit more expensive. The BMR is the lower cost of the two and uses proprietary magazines which are more like a normal small 22lr mag. It doesn't fit the 700 foot print, but it does allow for a 700 trigger which can make a big difference.
CZ's are great value's, however some of the newer one I think are over priced. The 452 would be preference as it has dual lugs vs the 455 and 457 single lugs. 452's are getting harder to find and starting to creep up to what I would consider too high for what they are.
All that being said, if you're budget would be pushed a bit more, then an Anschutz 64 type action would be my next preference. An older 54 based action is great, but probably runs out of your budget.
I haven't mention a Remington 40x or Winchester 52, but those would absolutely be a great rifle if you can find one in good shape. They can run anywhere from $1000 to the moon depending what they are.
I don't know about everyone on this thread, but two of us shoot Rimfire Benchrest and F-Class competitively, so our expectation of precision is probably a bit more than most.
For me, it would go like this:
1. Custom - $3k+
2A. Remington 40x - $1000+
2B. Winchester 52C, D, or E - $1000+
2C. Vudoo - $2kish to start
3. Anschutz - $1000 +/- for a nice 64 action
4. Bergara B14R - $850-900
5A. Bergara BMR - $600
5B. CZ $500-1000
5C. Tikka T1X
There are others, but if I was going to spend 1k, I would be looking for a Remigton 40x, Winchester 52 C,D, or E, Anschutz 64 based rifle or the B14R where I could change out stocks and triggers easier later on. The Anschutz Trigger should be pretty good to start. The Remington and Winchester would have to be right time/right place types of purchase, but they are out there.