They would have been very aware of the Charge of the Light Brigade and the Crimean War in general.

First off, the poem The Charge of the Light Brigade was very popular after 1854, and I can't imagine it would have escaped any of these men's ears. Whether or not it penetrated their psyche beyond pop-culture alone I cannot say. Reports from the Crimean War were widely reported in newspapers in the United States and were important discussion points for Army officers. The U.S. Army of the 1850's was primarily focused on siege warfare and engineering, and the massive sieges would have been of great interest to them.

We also know that the U.S. Army sent three observers to the Crimean War who would have certainly had first or second hand knowledge of the event. Later Union General George McClellan was one of these men, though only a Captain at the time. McClellan directly observed the Siege of Sevastopol and the location of the Charge of the Light Brigade. His report on the war was widely read and disseminated through the U.S. Army and would have certainly been read by any officer. He also brought back over 100 books written by European generals and tacticians, the knowledge of which was also transmitted through the U.S. Army. McClellan, like most other military officers, thought the Charge of the Light Brigade was foolish, though he did not understand why it failed. He did not understand the power of newer rifled field guns and instead focused on the need for newer cavalry tactics and equipment and better communication and leadership. This resulted in a new book on cavalry tactics by McClellan which still included light cavalry armed with sabers, and the design of a new saddle called the McClellan saddle, still in use today. Most European generals didn't seem to internalize the power of artillery either, and focused on poor command and communication for the charge's failure.

There are also some important differences between the cavalry charge at Balaclava and the infantry assault, not really a charge, at Gettysburg. The Light Brigade consisted of just over 600 light cavalry advancing against a very large field guns emplacement uphill from a mile away. (if memory serves, I am an American historian, not a European historian so apologies if I mess up some of the Crimean details.) Pickett's charge was 12,500 men advancing across a slightly shorter distance. Confederate leaders also believed that their battery fire earlier in the day had been effective in silencing Union guns prior to the advance, but it ended up being a ruse. Longstreet seems to have been wise to this plan and opposed the advance, but Lee believed the battery fire to have been effective and still ordered the advance. It could have worked had it been larger or better supported. It wasn't as terrible a plan as some people make seem like (especially considering other Civil War meat grinders), and had it succeeded we would be talking about the brilliant if costly maneuver today.

In short, they were very aware of the famous Charge, but didn't understand why it failed. Even if they had the infantry advance at Gettysburg was sufficiently different, and Confederates sufficiently fooled, that it would not have mattered if they had.

Edit: I also forgot to mention that Lee was in direct contact with French officers who were sending him copies of maps, movements, and fortifications as the Crimean War was going on. He was also an acquaintance with one of Napoleon's brothers, Jérôme-Napoléon Bonaparte. Lee was the superintendent of West Point during the Crimean war and would have likely made sure the cadets were informed of the latest information, though most of the men involved in the Civil War graduated by then, the U.S. Army officer corps was a tight nit place, and they would have been in discussion with each other and at a minimum knew each other from the Mexican-American war.