Dimming LED with speed controller
An "inelegant" 1st try:
Place one or more suitably current rated diodes in series with the strips.
Two or three may be OK. Four MAY be needed.   

Using eg 3 x silicon diodes in series (1N400x will probably be OK) will drop around 2V before substantial current flows. If this "almost works" try either a 4th diode in series and / or to N diodes add a resistor in parallel with the strips to conduct some of the controller current. 

The LEDs need non zero voltage to light but the resistor will draw current at any voltage. 

Note that the diodes will reduce the maximum current that the LED strips will 'see' and so MAY reduce max brightness slightly.
For 10W max ~=  1A load, try say 0.1A resistor current max.
Try say a R = V/I = 12/0.1 ~= 120 Ohm resistor to start.
Lower or higher may suit.
Resistor max dissipation = Vmax^2/R