358WCF,
My two 338-06s are both the standard version, so slightly less volume than your AI. That said, my first look at loading 15.5 in my 338-06 carbine indicates case capacity is not going to be a big issue.
Since it's a faster powder than H4350 (or at least I am assuming it is - am I mistaken?) we would be using lighter charges of RL-15.5 than H4350.
I have plenty of H4350 and can do a comparison for you. I can compare the visual bulk of two identical charges, or set my measure to a fixed volume and compare the weights of two identical volumes. How would you like me to proceed?

Regardless, I will probably load up my next round of 338-06 with RL-15.5 later tonight and can provide pictures similar to those above. My sense is bulkiness will be less of an issue in the 338-06 than the "Nine-Three."

Cheers,
Rex