Here's the post I would heed the most and it will keep you well under your budget:

"I changed out the 7H1 heads on a 72 LeMans I had, with some 1970, small valve, #16 heads. No port work, just a good valve job, and new guides. Took off the Performer intake with a Holley vacuum secondary carb, and installed an older factory Q-jet intake and carb. That engine ran a lot better than it did before. Actually impressed me for a 350. It still had a completely stock bottom end with stock cam."

You might consider a bit more cam to really improve things and if you went that direction a better matched torque converter. All this would keep you closer to 2K(800 for heads, 500 for carb and intake, 300 for cam and other items, 250 for torque converter).

While I can't disagree with the recommendation of Len Williams, I do think it would put pressure on your budget, take extra time, and bring up other issues for you(posi rearend, better gearing, etc.)