Accurate is relative to what you want to do with it. There are a lot of YouTube channels of people hunting iguanas in residential areas. They’re hitting crazy accurate shots on iguanas at quite a distance!
Check out Iguana Solutions. She uses different equipment for different distances.
Benjamin-Sheridan pump, 20 cal. or 5mm. Only pump it up enough to do the job, very quiet and accurate. However, it's not even close to some of the nicer guns mentioned above.
I use a Benjamin marauder daily in hangars and own an air venturi avenger for my personal. Both are well within your budget and awesome, accurate air rifles.
Get a pcp otherwise you will wish you did. FX makes great guns, but for things like pigeons a Benjamin Marauder is a good starter. Accurate and quiet out the box. Others you will want a supressor Donny FL makes the best I have found. Get a bunch of different pellets for it PCPs sometimes like one pellet type. If you plan to do outside shooting for say groundhogs and raccoons I would get an FX, but they cost what 3 marauders go for. I like semi auto myself and use a Huben K1 works well with slugs and pellets. I shoot max 100yards with slugs and 60 with pellets and keep groups about 1/2 dollar without trying. When looking at air rifles forget FPS and concentrate on FPE you want one that at least will do 30FPE + but with PCP most are easy to tune down.