How do I find the resistive energy deposition of a flat braided aluminum?
How you do this sum depends on who you are doing it for, and why.

There's nothing complicated about the action integral, once the form of I(t) is known. You're not going to stand about in a lightning storm with a current meter and a Franklin kite, so you need to get estimates of I(t) from somewhere.

If it's just for yourself, for interest, or you're protecting your shed, then you'll search the net for typical values for lightning currents and waveforms, and play around with the numbers from those.

It you're installing a lightning conductor to protect a building, then you'll look at your local building regulations, which will say something like 'the protective conductor must withstand a strike current of foo kA for bar uS without melting'. You need to use your local regulations for two good reasons. Not only does lightning storm severity varies around the world, but also these are building regulations, you have to do the sums the way the local regulators who are going to stamp your papers want them done.