The LCD is not what is special. The special part is the black elastomeric connector. They are sometimes used with LCDs, since most of the connections are very low current, a high-quality connection is not necessary.
Generally the backlight will remain steady all the time.
There is such a thing as color sequential LCDs that strobe backlight, but they are not common.
Here is a modern LCD backlight driver chip. The ripple voltage at the output is only a few tens of mV which will result in very little ripple current.
Don't be confused by the PWM mentions in that ...
Look at the front page of the data sheet: -
The internal switch that causes the inductor to "charge up" is limited to 250 mA for the TPS61041. This means that going "balls-out", the maximum power that can be taken from the 3.3 volt supply is 0.825 watts. Inefficiencies might mean only 0.75 watts is delivered to the output load.
You want to drive 40 mA at ...
I think in your situation the easiest way would simply be to use a 13 contacts to 13 contacts board, then a new 13 pin FFC cable.
Both can be bought from common suppliers. These two are specifically from amazon.
13 pin adapter board
13 pin FFC 150mm cable
While I'm pretty sure your pitch is 1.0mm....measure it yourself to be sure.
The LCD has the conductive tracks which are not much difficult to see with naked eyes. Try to flick it against the light.
The black connector are called zebra connectors. Really cheap and you need some pressure alone to mate the LCD with the zebra connector.
https://en.m.wikipedia.org/wiki/Elastomeric_connector for reference