You could cam it up, port the heads, edelbrock intake, and swap to a carb if you don't have emissions where you live. A mild cam like one or two small steps up from stock would still let you run stock heads and valves with new springs. I can't estimate how much power you'll get, but it will be a very nice bump. You'll really be able to feel it. Use a performer intake and a 500 cfm edelbrock 4-barrel and I'll bet you will even pick up a couple MPGs. That budget will probably be over $1000 but not much.
You could buy and install a 350 with your stock (reworked) TBI, but without a core you might be looking at more than $1500.
An LT1 can be bought for not much money, but unless you're installing it yourself, forget about it for a grand.
Try some more mechanical things. Instead of getting more engine, take a look at your chassis options. Even the "tow package" wagons had 2.73 rears. Find a 3.73, I guarantee you won't be disappointed. It sounds like a big difference and it is, but we Impala SS guys often run 4.10s and that's not too much. At least 3.42s, but no less. Your car has a 4L60 in it (or some prefer calling it a 700r4). It has the widest ratio spreads of any tranny, so whatever you do, don't kill torque. If you cam it up too far, you'll not be happy. Swap to a 2300-2400 stall converter and make your shifts a little firmer. A gear swap and tranny doctoring will make a world of difference in that car. If your car has 2.73 rear, right now adding power to the engine may actually hurt performance and streetability. Its all about a matched set and chevy made that one at the extreme edge of what I would call "matched."
If you want, post all of your RPO codes that start with "G", "M", and "F" and I'll tell you what tranny, rear, and suspension options you have so we have a good base from which to start.
Also, I don't wish to be contrary, but I advise strongly against the 160 stat in this engine. Its a common modification for LT1s but that is because they are reverse cooling. In standard flow cooling, the stat opens when the engine water reaches a certain temp like 190. It sends 190 degree water to the radiator to be cooled off and returns as cool water. In a reverse cooling engine like an LT1, a 190 stat opens up at 190 and sends water to the engine, not the radiator. That means that it starts as 190 degrees and gets hotter. It goes to the radiator and comes back as 190 degrees. Switching to a 160 stat keeps "normal" operating temps in an LT1. Using a 160 stat in a standard cooling system doesn't let the engine get hot enough and oil won't be able to burn off the acids, waters, and contaminants that it needs to and oil condition will suffer dramatically. Plugs will foul faster, heaters won't work as well, fuel mileage will drop, and catalytic convertors can clog. Cylinder washing/scoring and bearing failure have been extreme and rare results and performance won't be helped in TBIs. In fact, the PCM compensates for the cold temperature by richening the mixture. Closed loop operation is already richer than optimal, so power will drop. In LT1s, power increases because they use a Knock sensor. The cooler heads allow for more ignition lead. Many drivers switched to a 160 because their car overheated. This is not an overheat cure. It just postpones it. Heat increase in the coolant is caused when the heat being added to the water is greater than the cooling system can remove. Even if you take the thermostat out, it won't cure over heating. Something else is wrong if your car is overheating.
That's all... off my soapbox now