Look at the CM-2. It is a Russian gun made by Izhmash. There is quite a bit of information on it at rimfirecentral. Sourcing one is a bit sketchy right now, but you can buy one NIB for under $400. One place to check is stanleysproshop.com. The list only the Cadet model (junior size), but can usually order one.
General consensus at rimfirecentral is that the CM-2 will shoot every bit as good as an Annie or other European target rifle. You are giving up fit and finish, not functionality. As a comparison, I own a Baikal IZH-46M air pistol (it may or may not be made by the same company as the CM-2, but the accuracy v quality should be the same). Fit and finish are quite good, though I needed some aftermarket grips to make it fit better. It doesn't point quite as well as the Morini and Pardini guns my fellow shooters have let me try, but my Izzy cost me $270. There is a difference, but not a $1000 difference (at least not to try the sport out).
Another option is to buy a used gun. Thadscott.com has a bunch of older Anschutz rifles, in both the 64 action (good) or 54 action (best). They tend to be older and have more markings on them, but they should shoot.
Info on the CM-2:
Discussion of Anschutz v CM-2: