Here's what I would suggest given you're looking to swap the old 4 speed for a T5. Granted some of this will probably be easier said than done, but I think it'd give you what you're after. If you're looking to rebuild the engine and all of that, I'd just have the heads checked and refreshed. That should keep costs down some. Replace the pistons with hypereutectic flat tops (assuming you'll need to have the engine bored and replace them anyway....), find and buy a dual plane intake (one like I have would work great as it's basically an Edelbrock without the Eddy price), Replace the cam with one that's power band is somewhere in the 1500-4500 rpm range, add a 600 vacuum secondary carb and call it a day. It won't make blazing power but it should be noticeable compared to what you have now. In your budget I'd also suggest an air fuel ratio gauge so you can adjust the carburetor should it need it. I'm kinda speaking from experience with that since mine needs tuned in the worst way and I don't have one, so I guess I'm trying to teach from my mistake. In any event, a combo like that should yield you somewhere between 250-300 fly wheel horsepower and be driver friendly. While you're at it I'd say to swap the rear gears for some 3.55s. The T5 will allow the slightly steeper gear to give you the seat of the pants difference and still cruise on the highway without revving to the moon.
There are definitely other ways to go about it and others out there who know a lot more than I.
But keeping budget in mind and making it priority, and assuming you'll have the heads reworked, this would be my initial shopping list:
You already have headers so that's not something you'll have to try to locate and buy. Everything else can either be purchased at your discretion or your machinist may have connections to be able to hook you up with some components - specifically pistons if you're over-boring the block. Now, if you can get your heads drilled and tapped for screw in studs a set of roller rockers would be added to that list too, but that might also necessitate taller valve covers. And again, if it were me and that were done, I'd look to increase the ratio from 1.6 to 1.7 which effectively increases the valve lift. For the cam I listed the lift would go from .472/.496 to .502/.527. In that case you'd want to double check piston to valve clearance. You'd also need to double check what valve spring pressures and heights you would need.
As for the gear swap, you
might get lucky and find someone or someplace that has an 8 inch with 3.55 gears already. My gears were from a V6 II that I got as a parts car. If you can get that lucky, just swapping the carrier is a simple project that will yield a great seat of the pants result. If not then you're relegated to buying a new gear set. Swapping the gear set really isn't as hard as some think it is but you'll definitely need a few specialty tools to get the job done such as a dial indicator and a press. Adding a locking or limited slip differential wouldn't be a bad idea at all either, but not totally necessary if you're not really interested in stomping on the go pedal from a stop (like some of us with lead feet do from time to time....

).
And as already mentioned, the heads are basically the choke point of these engines. While they cost about as much as the combination of parts I already listed (or more!), they dictate what kind of power those components are capable of. A decent set of heads like the Twisted Wedges I have can bump the 250-300 flywheel horse power to the 300-350 range. They will quickly eat a budget, but if you can swing it they will be an improvement over production heads and yield more power in general. But, in that vein, since you're not looking for a high rpm screamer, you'll most likely want to stay with smaller ports, instead of the hogged out version of whatever head you were to choose.