I'm in agreement with Legate.
Even assuming that the sole difference was Sauron's continued presence in Dol Guldur, and that everything else (up to a point) was identical to the book (just for fun), there is no way the Fellowship would have gone near Lorien after the escape from Moria.
Mount Doom may have been more accessible (if Mordor wasn't infested with Nazgul and orcs), but one would have to be in the general vicinity of Mordor to be able to avail of that accessibility in a useful way. The Gap of Rohan would still have been closed, the route through Lorien (or between Lorien and Mirkwood) would have been madness, so that would have left either going around the White Mountains by west and south, or taking a similar route to Bilbo's through Mirkwood, then down the River Running and across land. Both would have taken too long, and who knows what mischief the Big S would have gotten up to in the meantime.