I am not a huge fan of trading for Lavine, but I think you find the minutes for Monk and Lavine by using an occasional three guard lineup, not that you start one or have it be your primary lineup. I'm guessing that you would probably see the three guard lineup most often in closing situations (maybe the last five minutes of the game), and go Fox-Monk-Lavine-Keegan-Sabonis. Minutes would be something like:
Fox (34) Keon (14)
Lavine (22) Monk (26)
Keegan (28) Duarte/JTA (12) Lavine (8)
Lyles (24) Sasha (16) Keegan (8)
Sabonis (34) Len (14)
That's 10 guys getting regular rotation minutes, with at least 4 of them (Keon, Len, Duarte/JTA, Keegan) being defensive minded and at least two others (Fox and Lyles) giving good effort on D. You would still have guys like Kessler, Colby, Duarte/JTA who could mix in for those 12 backup 3 minutes, and also Javale if something happened to Len