I use a NF NXS 5.5-22x56 and have used it on a few occasions to shoot pigs by moonlight without any other illumination, and it certainly is quite capable of visualizing deer during legal shooting hours under most all conditions. Definitely can recommend it and it should be available within your budget, especially a lightly used one. The ATACR's are even better, but physics is the law, go with a bigger objective, 50 or 56mm. S&B PMII's, especially the 3-12, are the other obvious choice, which might be available in that price range.
Professionally trained and certified concealed carry and license to carry instructor.