I am using a common negative switch for doing so
The relay and LED side is plain.
It's the controller description that does not make sense; "Common negative" implies it switches to ground??
The connections from it are all shown individually, without any common or power/ground info, so it's not clear what it is switching where or how??
Do you mean that is is intended for LEDs that have their common to negative (0V)? In other words, the controller switches the positive side?
If so, you could use appropriately rated transistors or MOSFETS, so they are switched on by the positive signal and pull the LED connection to near 0V.
Example, something like this, using a MOSFET - one of the LED colours being where the lamp is shown:
Rin limits the peak current gate charging current, and if the control voltage is higher than the required or maximum gate voltage, that and Rgs act as a divider to limit the voltage to a safe level.
Rgs ensures the MOSFET is off when there is no input.
For a typical power MOSFET the gate should be at around 10 - 12V for minimum dissipation when "ON".
If the control device produces a 5V signal, then you need "logic level" MOSFETs that can operate with lower gate voltages.