Carburator size

My 340 isn't that far off of your 360 build, specs are

'68 340, .060" over with KB 243 hypers, .018" over the deck, ~9.8:1
308 heads, 2.02/1.60 valves, 65 cc's chambers, stage II ported and flowing 264cfm at .500"
Harland Sharp 1.5 roller rockers, Smith pushrods
Lunati 60404 Hydraulic cam, .513/.533", 276/284 advertised duration
Doug's 453 headers
Edelbrock air gap rpm intake

Hp should be in the 420-430 range as well. I tried running it with a 670 Street Avenger, was way lean out of the box. I had a 770 street avenger on hand, so rather than start with a big tune on the 670 I put the 770 on it. Runs pretty well right out of the box with the 770, but if I change anything it'll be to go up a size on the primaries. Seems pretty close, but its definitely not rich as the plugs are still bone white. It's getting a wideband A/F gauge here in the next few days so I'll have a better idea of where I'm at.

With your set up I don't think a 670 is going to cut it.