Flickering LCD backlight with fingerprint scanner
The backlights on these LCDs are just an LED array with a series resistor so the intensity will be directly proportional to the input voltage; flickering voltage = flickering backlight.

Check these things:


Measure the output of the 5V regulator with an AC volt meter to see if the regulator is oscillating when the backlight flickers, i.e. AC voltage >> 0V.  If it is, then you probably need to increase the size of the output and possibly the input capacitors.
Measure the Input-to-Output voltage differential of the regulator and multiply it by the output current.  If you dissipate more power than the thermal design allows this will heat the regulator making (1) more likely as well as driving the regulator into thermal shutdown.


If the regulator is linear, i.e. 7805 etc, you can solve both (1) and (2) by changing the regulator to a drop in switching type.  These will dissipate heat as a percentage of the output power instead of as a function of the input voltage and can be made self contained with regard to input/output capacitors which would be on-board instead of off-board.  A switcher can also do something a linear can't do:  Supply more current than the source.  A 12V 0.5A power supply into a 5V regulator can supply 5V at 0.5A with 3.5 watts of waste heat.  That same 12V supply into a decent switcher can deliver 1.14A at 5V with only 300mW of waste heat!

This PDF is an example circuit that runs about 95% efficient and fits on a 7805 footprint.

This PDF is the circuit of the 1516 switcher linked above configured to replace the NCP1117 linear regulator use on the Uno Rev3.

This PDF is a full thermal analysis of the Arduino Uno NCP1117 regulator running under the conditions stated in your question.  As you can see, you were right on the edge of what the NCP1117 could handle at 200mA (LCD plus scanner).  I have included a thermograph showing the same unit supplying 1.85A at 5V using a drop in switching regulator with no issues.