The interior support beams of my trailer aren't solid. They are aluminum and are shaped more like C-purlin, with the opening towards the inside of the trailer. I wasn't sure if it was going to work or not, but I just tried attaching the first screw to put up a furring strip and it just isn't going to hold.

What else can I do?

What is not holding? What brand trailer are you doing?

If the C section of the channel is open towards the inside of the trailer, a strip of wood could be ripped to fit inside the channel. By fastening it with screws through the sides of the channel, you will now have a permanent furring strip of which to attach to with your interior panels.

Gard

Good idea Gard. Will be a pain since I already insulated those little spaces and taped it all up, but it may be my only option.

It is a 2007 Elite trailer.

Hubby says he thinks I was too 'sloppy' while trying to drill the self tapping screws into the aluminum. I am 98% sure it still will not hold for him.

Well crap. Hubby was right. Now that I had one screw (loosely) holding things in place (instead of trying to hold it, stand on a ladder and drill) I was able to drill better and the screw I put through held everything tightly in place (the other hole was "wallered out" - hubby's terms).