When you say "no depth left for screws," do you mean there's no material to put the screws into, or that you can't have screw heads sticking up into the channel? If it's the latter, just counter-sink screws so the head is flush in the channel. Otherwise, a few dots of adhesive caulk along the channel should hold it well.
If the channel is routed already, the adhesive tape I linked won't work because it will add thickness and you won't get your flush finish.
Can you link the LED units you plan to use?