Your translation is fine in that it gives meaning and is understandable. I put a few corrections, though.
I'd translate the sentence as below.
식물 위주의 미래를 선택하라. 축산물은 필요없다. (an imperative sentence with strong and forceful tone; suitable for a title of an article)
식물에 기초한 미래를 선택하십시오. 우리는 육제품이 필요 없습니다. (an imperative sentence with mild and persuasive tone; suitable for a casual campaign)
Do you think "식물 위주의 미래를 선택하라. 축산물은 필요없다." could be used on a protest placard - in that sort of advocacy context? It is fine if the language is stronger and more direct, since this is so important. I also like that this translation is shorter than the other one.
Also, in the second sentence of "식물 위주의 미래를 선택하라. 축산물은 필요없다.", do "animal products" become "livestock products"? Or is this nothing to be worried about? The language used by the industry is part of the problem, and, in English, animals often get reduced to a stock that just so happens to be alive. "Animal product" on Wikipedia is the meaning I would like to go for.
I thought 축산물 and 육제품 mean the same, but I found out that they are not the same.
육제품 simply means the products made of animal meat and intestines.
축산물, however, indicates a broader range of 육제품. The term, 축산물, includes raw materials used to raise the animals, processed foods of the animals, any materialized left-overs of the animals, such as skin, horns, fur, etc.
So, if you are referring just the animal meat, you may use 육제품. If you are referring the whole thing of processing animal products, you may use 축산물.