Looks like there is to many trees. Im no gardening expert (im rough as guts) but unless they are gone your new lawn is going to be dead in winter from the leaf drop. Weeds need cutting to dont forget!
Will you be getting a monthly maintenance contract or is it just a one off job? And yes you should already know this.
If its a monthly maintenance contract then id be saying "its a waste of your hard earned doing a big make over, let me spend a little time to make it low maintenance for you and the agent and then ill look after it from there at $? per month. That sounds pretty good doesn't it?"
If its not then yea, get as much as you think you can. Its winter.
Ive found skips fill up to fast (especially at night..) and id rather the money for the rental come into my pocket.