You will get whatever thrust is required to get and maintain 1000 to 2000 fpm rate of climb.
I can't find anything to say that it will provide whatever thrust is needed to achieve it. I used to assume that until I read BOAC's post and then I started to look. It seems to me that 1,000 - 2,000 fpm is a pretty wide margin. If somebody said to me that the Boeing engineers figured out a narrow reduced thrust range that would achieve that for all normal conditions then I could accept. Also, it says somewhere that it will maintain pitch until RoC increases and then maintain speed, which also suggests a fixed (or near enough) thrust.
I don't know and really Boeing seem to have decided that we don't need to know how it achieves it, just what to expect and what to do if it doesn't happen.
However, clearly then manual thrust less than GA thrust is not unknown so long as you are achieving 1,000 fpm! Personally, I just push them forward and then get the PM to set it to the bug.