[...]
>
> I.e.,it seems that Org is going to do its own "read.table" before even
> looking at the code in the source block.
>

Yes, this is true, Org will use read.table to read in tabular data.  See
the code in lisp/ob-R.el for specifics.

>
> Is there some way to get Org to use the "fill=TRUE" option on a case-by-case
> basis?
>

Yes, The attached patch allows the :fill header argument to be specified
adding "fill=TRUE" to the read.table function call.  Please try it out
and let me know if it works for you.