So now I have a rat problem... Which pellet gun should I go with? Nv or maybe IR

A springer or gas piston destroy will a nightvision in short order, how many rats ? a beeman 78 is good rifle in c02 

youll get quite a bit of rat poison for 250....

also rats dont take a lot of power to kill, wadcutters to the head and you could do it with a hw35. 

Like the others said, rats aren't hard to kill. The UK guys do it all the time with sub 12ft-lbs rifles. If you wanted to go for quiet you could pick up a Benjamin Discovery .22 and use Co2 in it along with a TKO muzzle brake to handle any excess noise. Of course the beauty is if you wanted to do HPA you could at a later date.

As for sighting you could coble together a IR Scope cam setup like some of us have done. One I use gets me to about 50 to 60 yards with a 850nm IR light. However a red LED light should be able to work just fine in spotting the rats without them spooking to badly. Personally just try a Red light clamped to the scope and try a trail run on the rats to see how it all fairs.

Crosman 1077 ?

I asked about low cost ratting on the Planet Airgun forum and suggested using solar powered lights. 

http://planetairgun.com/index.php?topic=8667.0

You need to find out what is drawing the rats to your yard...what are they eating? The world has an endless supply of rats and although it is fun to shoot them you cannot eliminate them that way if something continues to draw them to your yard.

This was the NV system that I built:

If you see 3 or 4 that means you have 3 or 4 hundred. Two years ago I saw 3 rats during the day..... Now I have a body count at over 500.

Are they rats? Or are they Voles? Not moles but voles. Voles look like mice but with a smaller tail. They eat vegetation. They use mole holes most of the time. They like high grass/weeds for cover. I have them by the hundreds. You can't get rid of them completely either.

Get a few cats! That ought to do it! Other than that, if its not too far of a shot, like someone mentioned, a 1077 for close range due to the amount of lead that can be slung quickly, maybe a long barreled S&W 586 or 686 or the full sized Desert Eagle which are the most accurate action pistols which have higher than the norm velocities which can shoot as quickly as you pull the trigger (or hammer). Bait the area to draw them out in strategic locations relative to yours. Peanut butter, cereal, meat, fish, dried shrimp can be used to experimented with.
I am not sure night vision optics can last when mounted atop a spring or gas ram gun. I would suggest a Beeman R7 or HW30 both in 177 out to 35 yards. Quiet, accurate, easy to cock and shoot with less fatigue than a magnum powered rifle.
If you can find a Benjamin Discovery or Maximus for a relatively cheap price used with a pump, you just may get lucky to find a combo that will fit your budget, throw on a TKO and its pretty darn quiet. They pump up to only 2000psi max so it would be fairly simple to fill with a decent amount of shots which could give you good accuracy out to 45 yards and the night vision equipment would be safe too. A modded 1377 or 1322 with upgraded steel breech with a flat top piston and hot valve with carbine stock and 18 inch (or longer) barrel with a TKO at the end of it would do just fine and within your budget out to 40 yards. I would personally go for the 1377 (177) higher velocity for flatter trajectory for longer range shooting. With either one of these pumpers with TKO, pretty darn quiet.

Here you go. https://smile.amazon.com/dp/B00NW53Z2U/ref=wl_it_dp_o_pC_nS_ttl?_encoding=UTF8&colid=DRQQN9EWL9F9&coliid=I2E20FFKB84V1E

Just pick up a mount to put on your scope. Several listed on the same page. Any decent break barrel will get the job done. I like Hatsans.

Ed