SONAR X3 and HDMI out

Hello all:
 
I've taken a small part-time position teaching guitar classes, and while the students walk in, I wish to play over backing tracks. Unfortunately, I'm not given a mixer board, but an HDMI connection to a TV. Right now I play the tracks (either midi or audio) on the laptop and bring a Roland Microcube. It sort of works, but...I'm already schlepping a briefcase w/laptop and papers, a guitar, and a stand. The Microcube requires an additional backpack, and I look and feel like a sherpa.
 
I thought OK, I'll bring the laptop and my Scarlett 2i2 and stuff that into my gig bag. Well, SONAR does not see the HDMI out, only the Scarlett outs, so I cannot run my tracks through SONAR and use the wonderful TH2 Overloud thingie.
 
I have an iPad and bought an HDMI adapter. I fired up Amplitube and ran it with iReal Pro tracks and it was horribly latent. Coming out the 1/8" out via the iRig it's useable, but as I said, I only have an HDMI connection.
 
Behringer makes this cable with an interface in it--1/4" to USB, but no outputs (it's just a cable). If I used that, will the HDMI out play nicely with it?
 
Finally the A in AV stands for AUDIO, why does it have to be so different setup-wise than musical audio equipment?
 
Or more simply, how can I get SONAR to see the HDMI?

I didn't explain myself very well, I'll try again. I can get the tracks to play, but I want my guitar coming through the same source both for sound reasons and I don't want to bring an amp. I want to play the guitar through the computer while backing tracks are playing. I have a Scarlett 2i2 interface which will fit in the pocket of the gigbag.
 
Or is there some other way to connect to the computer? If SONAR would see the HDMI I could assign the output to that and the input to the Scarlett and I'd be golden. I just installed Amplitube, and it asks what technology to use, and if I pick ASIO, no other options are available as far as I/O goes. If I don't choose ASIO I cannot use the Scarlett and then I'd be dealing with the 1/8" input to the laptop (always fun). I would do that if I had to, I just don't expect good performance (i.e., extreme latency would likely occur).