The Full choke barrels are safe with steel shot but I would not use sizes larger than #3 steel as there is a possibility of bulging the barrel. It is unlikely to burst but the bulge will be noticeable to the naked eye. Patterns also tend to be rather poor when one uses more choke than called for.
The same can be said for the other harder than lead alternatives though bismuth, tungsten/poly, and a few of the other shot types would be perfectly safe.
Your Rem-choke barrel would be just fine for non-toxic shot though I would hesitate to use the Full choke as it could be damaged if using large shot. In any event, I have not found Full to be condusive for best patterns with large shot so wouldn't consider it to be a factor.
+1
I would get the long barrel tubed, personally.
