I did a little research...
First off, here's a blurb from part 91:
Sec. 91.3 Responsibility and authority of the pilot in command.
(a) The pilot in command of an aircraft is directly responsible for,
and is the final authority as to, the operation of that aircraft.
(b) In an in-flight emergency requiring immediate action, the pilot
in command may deviate from any rule of this part to the extent required
to meet that emergency.
(c) Each pilot in command who deviates from a rule under paragraph
(b) of this section shall, upon the request of the Administrator, send a
written report of that deviation to the Administrator.
Second, the "Climb to 1000 then climbing turn to 5000" in the MAP is there for one of two reasons:
(a) There is an obstacle in the direction of turn that requires 1000' MSL to clear it safely, or
(b) The people who live in the direction of turn have bugged their congressperson about aircraft noise.
Checking the pubs, there are no "Non-standard Departure Minimums" for 36L at KMEM. This should give you the warm and fuzzy that your transport certified jet can climb above any nearby obstacle with the critical engine inoperative, especially since you are performing an approach climb (climb started at approach end, not departure end). This means that the 1000' prior to turn is for Joe/Jane voter, who I say can sc*&w themselves when an airplane is in trouble. For your concern about turning ABOVE 1000' instead of AT 1000', I'd refer any jacka*$ complaining about that to Part 91.3(b) as above, and then tell them you will write the letter explaining yourself to comply with Part 91.3(c).
My $0.02.