As has been mentioned, the problem you've run into is that AMC changed Auto tranny designs in '72. Earlier used a Borg Warner, and later used Chryco. Unfortunately, the back end of the crank is different for each...
To overcome this, there are two types of adapters on the market. The first, designed by AMC, requires machining of the crank, making switching back impossible, not a good idea in your case, when you want to return the 360 to the Machine! The second version came out later, and does not require machining of the crank. Talk to the AMC vendors for price, they'll know what you're talking about!
As for balancing, there's two way's of doing this as well! The first, and best type, involves tearing down the engine. The Pistons are weighed and matched, the Rods are weighed and matched, then weights, based on the pistons and rods, are attached to the crank, and the crank/damper/flywheel are balanced. That's the best way, but obviously it's more work! Next best is to match components, often referred to as match balancing. In this case, you'd take in two flexplates, and ask the shop to make the new one match the old one. In your case, where you're planing on putting the motor back in the Machine, match balancing makes the most sense.
I don't like suggesting you spend money needlessly, but I have seen emails from respected AMC people, like Tony Zamisch, stating he's seen differences in early/late 360 flywheel counter weights, and balancing is a problem I've personally hit, with my '70 390 (that I bought and later discovered had a 258 I-6 Flywheel!) I'd hate to have you shake the bearings out of your newly rebuilt 360, while running it in a temporary body....