You could use swing set chain if you wanted. You just need enough of it to hold the weight you are hauling.
3/8 is a pain because it is heavy. I would get some 5/16 and if you ever need to haul something real heavy just use extra chains.
A chain needs to be long enough to go from point A to point B. Only you can tell us how long that is.
The most important thing to remember is your chain and binders must be marked.
Lets take you 5/16 4700 lb chain. That is grade 70 transport chain. As long as you can still read the markings that tell you this is that type of chain you are good to 4700 lbs.
But lets say the tag falls off; the stamping gets rusted over or the paint fades.(depending on how it is marked) your 5/16 4700 lb chain is now only good for 1900 lbs grade 30 standards.
Even though your chain will hold much more if you can not prove what grade chain you have it will be considered grade 30.