My goodness no. Really DON'T.
Not only is it dreadful, it's so far removed from your other DC names that it comes across as really unfair - lovely classic names with good shortenings for the others and then this-?!
Agree that your DH needs to either buck his ideas up or face that fact that you'll be naming this one! 'It's the only name I like' - no sorry, not allowed, that's code for 'I want my own way with no compromise'. You have three others who he presumably managed to come to reasonable agreement on? He goes away and compiles a list of five like every other reasonable person, sorry.
Also, never never never choose a name based on the fact that you want a certain nickname, and especially not on a whim like this which results in a dreadful made up name. Leveson because he likes Lev? So what does he do when his DS and peers decide he's Lee? Or Sonny? Um...
Lev is a bit random but not awful. Go for Levi if that's what you want, but again, this jars with your other choices.
The one that springs to mind is Sebastian, similarly long and classic with the short form Seb, which isn't a million miles from Lev in its 'feel'.