Okay I finally got to the bottom of this.
First I reverse engineered the circuits: http://imgur.com/a/24TFn
Then I found the datasheet for the backlight driver chip: http://www.onsemi.com/pub_link/Collateral/CAT4139-D.PDF
Marking code indicates it is the same chip in V1.0 and V1.1. There are two differences between the boards:
1. The V1.0 attempts to implement "PWM on SHDN pin", and the V1.1 implements "filtered PWM". The difference between those modes is explained on page 10 of the datasheet.
2. The V1.0 is incorrectly laid out, and the VIN and SHDN pins have been reversed. That means the PWM signal is incorrectly fed into the voltage input instead of SHDN, while the SHDN line that is supposed to be PWM'd is connected directly to VCC. This is bad and to be honest I don't understand how the thing even works. It must be drawing parasitic power through the SHDN pin. It is almost certainly why the PWM has been disabled in software.
As for telling you how to force the V1.0 to PWM. I don't think that is a good idea given the layout error.