I've got a nice recording of Lotte Lenya singing the Blitzstein lyrics, and she doesn't have any trouble singing them to the tune. As to how well the MH lyrics can be sung, the syllables don't scan properly in many places, the language is often awkward, and obscure where it should be direct--lines like this following don't have much punch:

"And the ghastly fire in Soho-
Seven children at a go"-

The thing is, these are theatrical lyrics, the audience neither knows nor cares how closely the language tracks the original German--they don't even care that there was an original German--they respond only if the show speaks to them--