How to handle unprofessional negative reviews from a journal?
It appears that the editor gave more weight to the first review, or at least gave you the "benefit of the doubt". Since you have the chance to re-submit, and you decided to do so, you should have to respond point-by-point to every comment that both reviewers made. Now "respond" is a deliciously diplomatic word that includes all the spectrum of reactions: from, say, "I did exactly what you asked" to, say, "go find a hole to hide and die, you stupid ignorant".  

The main advise I have to give is : keep your judgements to yourself (about which review was helpful to you, whether the one appears sloppy etc), don't differentiate between the two reviews in your response, and don't be more aggressive in your writing style when you respond to the points made by the second.

For example, Don't defend not comparing your work to concept X because "it is completely out of date". Defend your choice because, say, "concept X, valuable indeed when it first appeared in the literature, has been succeeded by etc see reference 1, reference 2, reference 3". You get the spirit. If the "outofdateness" is your opinion and the field does not agree with you,then, you have another paper in the making, one where you will try to convince the other scholars of your field that concept X is indeed outdated. 

If you do that correctly, the editor will certainly get the message (and well corroborated) about what you think of the quality of the two reviews, and why, even though nowhere in your response such an evaluation will explicitly appear.