Your suggestion about a mild 390 GT build up
Haven't fussed with a 390 in years, but, the one thing you want to think about really hard is gasoline.
A good modern cam grind can make a lot of power with a lower static cr, and still be good to drive.
The big Ford manifolds won't clear your shock towers, or I'd sell you mine!
What I recall from my '67 GT is that is ran like a scalded rat to about 4800 or so, but the small carb and corked up exhaust kind of flattened it out.
To really run hard, it needs more carb and exhaust (cam's really pretty good) and MUCH better valvetrain. The GT's had non-adjustable rockers and valvespring that were a built in rev limiter.
I'll let someone who's built one in this century take the wheel here. But if you had the dough, it would be hard to beat the Edelbrock Performer RPM system.
The only things for sure to do:
Pick up a set of CJ nut-and-bolt rods, they're stronger than the S- or Z-code stockers, but not much heavier. Get the CJ oil filter adapter and match your block openings to the ports, you get better flow thru the filter. Use a windage tray and the 7-quart 427 pan, that keeps the oil down where it belongs. Get the adjustable rockers and use them with some good valvesprings.
If you're bucks up, you can stroke it to 410 with a CJ crank, or one of the aftermarket strokers. Just make sure all your recip weights are right and the block's not too thin in the walls ( the greater thrust angle can find a soft hole every time).
A 390 will probably never be the king of the street, but you can run anyone hard to 90-100 mph with one. And they look really cool all dressed out.