The passage in question is as follows:
But everywhere he looked he saw the signs of war. The Misty Mountains were crawling like anthills: orcs were issuing out of a thousand holes. Under the boughs of Mirkwood there was deadly strife of Elves and Men and fell beasts. The land of the Beornings was aflame; a cloud was over Moria; smoke rose on the borders of Lórien.
It's notable that there are three conflicts mentioned in this passage:
- One in Mirkwood.
- One in the land of the Beornings.
- One on the borders of Lórien.
If we cross-reference this with the Tale of Years (Appendix B) we see that the Breaking of the Fellowship happened on the 26th of February. But yet:
- The First Assault on Lórien didn't occur until the 11th of March.
- The Battle under the Trees in Mirkwood didn't occur until the 15th of March (same day as the Second Assault on Lórien).
The conclusion from this is inescapable:
- What Frodo is seeing from Amon Hen are actually events in the future.
- These events need not necessarily have been on the same day.
Although there's no record of an attack on the land of the Beornings in the Tale of Years, if we assume that what Frodo was seeing was the First Assault on Lórien and the Battle under the Trees, and we remember that these were on different days, then there's no reason why the attack on the land of the Beornings couldn't have also been on a different day.
Let's assume that what Frodo saw was the Second Assault on Lórien and the Battle under the Trees. There's no doubt that forces from Dol Guldur were split at least two ways, first against Thranduil's realm, as is confirmed by the Tale of Years:
Battle under the trees in Mirkwood; Thranduil repels the forces of Dol Guldur.
And second against Lórien (Tale of Years again):
Three times Lórien had been assailed from Dol Guldur, but besides the valour of the elven people of that land, the power that dwelt there was too great for any to overcome, unless Sauron had come there himself.
We know that the Battle under the Trees occurred on the same day as the Second Assault on Lórien - 15th March - so there's no doubt that forces from Dol Guldur were split on the same day.
Now, Lórien and Thranduil's realm are quite far apart (about the same distance as Lórien is from the Shire according to a rough measurement on the LotR map), so there's no doubt that these were two separate armies.
So the point is: if Dol Guldur's force could be split in two, I fail to see any reason why it could not also be split in three.