What you want is a camera with the smallest f-number and the largest sensor. DSLRs do well, but that would be hard to do on your budget. I could possibly recommend some used DSLR stuff but it would be right at the edge of your budget and not be available everywhere.
These will all be a few years old so that they are in your budget.
For a single all-in-one, look at the Panasonic Lumix DMC-LX100. It has a very large sensor (this one is called m4/3); it splits the difference between a DSLR with an APS-C sensor and the typical high end compact with a 1" sensor. New it is just over 600, used for 450. At the wide end of the zoom range it is f/1.7
Close but not as good is the Canon PowerShot G1 X Mark II. It as a microscopically...