The Holland lop rabbit breed is from 1952 and was only introduced to America at the end of the 1970s. That's what it says on the Wikipedia page for that breed already.
Rabbits are also certainly not the only species known for having droopy ears. These are found for example in pigs, horses and asses. That is just a quite common sign in many domesticated species.
Further, speculation of cowardice of German American units in that battle giving rise to any epithet is baseless. The Bull Run battle was in 1861, but we see the slur was already well in use by then.
Like for example in 1854, fittingly under the subheadline hypocrisy and tyranny:
a large majority […] are "ignorant Irish and lop eared Dutch."
And for the insulting accusation of "cowardice":
Commonly referred to as "Dutchmen" by other Union soldiers, and "lop-eared Dutch" by Confederates, German-American units in general earned a reputation for discipline. Some of them had previously served in European armies, and they brought valuable experience to the Union Army.
Wikipedia: German Americans in the American Civil War
So, the real reason for using an ethnic slur is found in the realm of nativist attitudes and political expression:
Historians may have erred in their interpretation of the election of 1860. If they have, they were led astray by the voluminous contemporary documents which relate the enthusiasm of German liberals for Abraham Lincoln, by the active part taken by prominent German speakers and writers like Carl Schurz, Gustave Koerner, George Schneider, editor of the Illinois Staats-Zeitung and by the numerous resolutions passed by Turnvereins andthe military displays indulged in by these gymnastic societies.
This activity may have been only the noisy demonstration of a small minority, but certainly the contemporary newspapers are filled with notices which give the impression that the Germans were vitally interested in the election.
Democratic vituperation against the "lop-eared Dutch" has helped strengthen the assumption that German sympathies were Republican. In 1860 the German-born residents of the northern states amounted to 5.74% of the population a small percentage but sufficient to be a deciding factor in doubtful elections. Republican politicians saw to it that their newspapers constantly wrote "leaders" to flatter the Germans, who were presumed to be a liberty-loving people escaped from the tyranny of European monarchies.
This impression was cultivated in the minds of contemporary Republican politicians - as well as later historians- by the German American press. When the execution of John Brown was announced with mourning borders by Der Demokrat in Davenport, Iowa, it is easy to understand how party managers might have inferred that the German population was inclined toward abolitionism. So too, when the German American politicians convened in Chicago immediately before the national Republican convention it is not surprising that the Republicans adopted the Germans' resolutions as planks of the Republican Party.
Jay Monaghan: Did Abraham Lincoln Receive the Illinois German Vote?
Journal of the Illinois State Historical Society (1908-1984), Vol. 35, No. 2 (Jun., 1942), pp. 133-139
And that brings us towards Missouri, the compromise state, where attitudes were most divisive.
By 1860, Missouri's initial southern settlers had been supplanted with a more diversified non-slave-holding population, including former northerners, particularly German and Irish immigrants. With war seeming inevitable, Missouri hoped to stay out of the conflict by remaining a part of the Union but militarily neutral – not sending men or supplies to either side and pledging to fight troops from either side who entered the state.
The policy was first put forth in 1860 by outgoing Governor Robert Marcellus Stewart, who had Northern leanings. It was reaffirmed by incoming Governor Claiborne Fox Jackson, who had Southern leanings. Jackson, however, stated in his inaugural address that in case of federal "coercion" of southern states, Missouri should support and defend her "sister southern states". A Constitutional Convention to discuss secession was convened with Sterling Price presiding. The delegates voted to stay in the Union and supported the neutrality position.
Wikipedia: Missouri in the American Civil War
As we see in this letter:
This is a mistake one of us made. The term jayhawkers was applied by Missouri rebels to all Federal soldiers coming from Kansas, while those coming from Springfield, Missouri were called by the same people Lop-Eared Dutch [German Union Regiments]. We soon got so that we did not care for it. Nor do we now, the names are still applied by the Bushwhacking Class. [Even though the Civil War ended in April of 1865, the animosity and bitterness that the war created between the loyal Union and Confederate citizens of Missouri and the citizens who lived in Western Missouri and Eastern Kansas continued well into the 20th Century.]
Arnold W. Schofield: Battlefield Dispatches, Jayhawkers and Bushwhackers, Nevada Daily Mail, Sunday, March 12, 2006
Thus we see an immense hatred towards those newcomers as such, and still 'un-Americans', among those who fought them. Coming to America, coming to Missouri, and then having a Northern, Union, abolitionist tendency? Not good in the eyes of those who came first to Missouri with their slaves.
For an example with many not nice words that would be censored here Colonel Monks and Wife A HISTORY
Southern Missouri and Northern Arkansas Being an Account of the Early Settlements,
the Civil War, the Ku-Klux, and Times of Peace.
By William Monks, 19O7
Missouri was a maelstrom of influences and nothing was as clear cut as our history books imply it is. The majority of people in Missouri were pro-Union but had to deal with a governor who was pro-Confederate. A number of Germans had settled in Missouri, centering on St. Louis. Their primary language was German but they were pro-Union. General Franz Sigel, an immigrant from Prussia, raised a number of Union regiments in Missouri and led them as a general. I fights mit Sigel was their slogan. The First Missouri, which found itself at the Battle of Wilson’s Creek was one.
The Confederates saw them as interloping foreigners who had no right to be in this family argument.
David Cary, on quora: "Why do Confederates refer to "Union men" as "Lopeared Dutch" in "A History of Southern Missouri and Northern Arkansas" by William Monks?"
Unionists simply called German recruits Dutch, but Confederates added the slur. It means just drooping ears.. And is meant derogatorily descriptive (Example1).
Give me the pail, you lop-eared buffoon. (p294, from 1854)
It could have been crooked nose, or pimple-faced – well, of in fact not the latter, if 'accuracy' would play any role in those insults, as those German recruits to the Union army tended to be a lot older than native born soldiers. But insults don't need to be accurate at all. "Ugly Dutch" or any other negative adjective works just as well. And that all union soldiers were cowards we just know as a standing stereotype among all those gallant Southerners.
Anti-German sentiment in the Trans-Mississippi.
"I Goes to Fight mit Siegel": Missouri's Germans and the Civil War Video Transcript