318 Senior Project Build

I'm trying to keep it around $1000-$2000 if that is reasonable for a rebuild. If I don't get the engine bored would I still be able to achieve close to 300hp?

You will likely come in at the top end of your budget for what you are planning...it won't be tough to hit the $2k mark with the cost of parts and machining.

I don't want to discourage you, but check into the cost of the machine work before committing to this. The better option may be the one that UOP gave...the 5.9 Magnum may be a better starting point than the 318 that you have.