What I've been able to gather about Monroe, is that when not fertilized, it'll hold fruit well into most of January (or beyond?).
As for Lula's that have not been fertilized, I picked the last Lula off a tree this year on March 12 (the exact date is on another thread: Mango and/or Avocado 24/7).
If you can, get'm both in a avocado cocktail tree (act), get 'Monroe' and 'Lula.' If you want only one tree, graft a budwood of one cultivar onto the other cultivar.