Did Milton Know and Respond to Lanyer's Work?

No "smoking gun" evidence exists as far as I know, but you might find some.  I'll accept a brief case for plausibility that M knew L because he read so much.  See his scholarly biography:

Author Wilson, A. N., 1950-
Title The life of John Milton / A.N. Wilson
Pub. info. Oxford [Oxfordshire] ; New York : Oxford University Press, 1983
LOCATION CALL NO. STATUS
  Main Collection  826.3 M66Swi