Transmission cooling line fitting size help

OEM replacement lines from the tranny (right stuff) and had planned to run rubber hose between the tranny cooler/radiator/tranny line.

Some of the tranny cooler kits that I looked out - shipped with straight rubber hose. Now, I am reading that RRR isn't a fan of that option. Should it be braided?

The brass fitting is out of the radiator - which I would like to replace with a different fitting. So one side is 1/8" pipe and the other 5/16 or 3/8s flare

Thanks, still learning