First, congratulations on your choice of FE for the Cobra. It's the natural way to go, with possibly the most beautiful engine ever out of Dearborn under the hood as a result.
If budget and streetability is your consideration, the 360/390 block with 0.040 overbore and
oil mods, Edelbrock, 427, or CJ-valved 406 heads, hyperuetectic or forged pistons with compression ratio between 10.5-10.95:1, a mid-performance cam with SOLID lifters (flat or roller) with adjustable rockers (check out CamResearch cams... they can pre-run-in the cam for you as well as custom grind for your specific needs!), a 390 crankshaft and stock rods, Edelbrock Performer 390 intake and Holley 600 carb, stock distributor with Pertronix module... it is just a great performer, totally kicks Corvette A*s, and is reasonable to build. AND SOUNDS AWESOME with Spintech mufflers... as good sounding as a 427.
I fully understand the pride thing. If that is a close second to budget, then by all means install a stroker kit, additionally, to get you to 427ci. NOBODY among us FE buffs is going to give you any guff for stroking your way to 427ci rather than boring. It's ALL good. The engine will rev a bit slower than the 390/406/427 3.78 stroke, but give you great 428 torque as a tradeoff. But the rest of the build can be budget as outlined, as long as you blueprint and balance the engine carefully. Use an excellent machine shop, not a lowbuck Bowtie Bozo.