Why does Sonar dropout when drumming nearby

Well two things may be a possible cause. I wouldn't use two sound cards for outputting. If the drums are close as you suggest it maybe jarring the laptop too much.
 
  

I want to make it clear that this is strictly playing two audio tracks at once, with absolutely no recording. One track is directed to the PA for the audience to hear. The other track goes to the in-ear monitors.
 
I know it's best to use one sound card with two stereo outputs,  but my only usb interface is this one with only one stereo output. So I'm having to use the internal sound card for the click track that goes to the in-ear monitors. 
 
It won't work to have the laptop in a different room,  as the drummer has to have access to it to control it. This is a pretty common set up in live settings, so I'm not sure what the problem is. 

I bet the band is loud! LOL
 
Be careful putting foam and things under a laptop. Laptops often depend on air flow they get from under and blocking it off may cause overheating.
 
John's idea of streaming the tracks from a USB stick sounds solid to me. It is a pity that all the music out front has to be mono though. That is a serious compromise in my opinion.
 
Perhaps some sort of multitrack hardware playback device that is designed to handle live situations might be an option.
 
 

The table/level on which the laptop is positioned is crucially important, too. If the table is very
feeble, it can resonate so much it's actually shaking. Also, the laptop should not be in front of the kick or a (loud/bass) loudspeaker. The sheer airwave shock can disturb the HDD.
Put bricks first, then foam, then chipwood plate, and if necessary, still pieces of foam under each corner of the laptop. And, as mentioded, do not forget ventilation.

I have played several hundred shows with a laptop in the same setting. And I've started out simmilarily to what your setup is. I was using a mid-range laptop for this but there was two things essential to make it work when the environment of the laptop gets LOUD (and possibly hot):
 
a) One soundcard for ALL the audio. For some reason the machine easily got a hickup when running anything from the internal sound card. Sonar has some serius issues with this. Live may give you better results, however, two soundcards is not ideal in any setting.
 
b) SSD or a very fast harddrive with big and fast buffers. I've used both and they worked well. SSD is the only sollution that I could rely on 100%. The USB memory stick thing sounds interesting and I think we tried it once. If that works it's excellent, but you might still have a problem on a stage with big subs and vibrations.
 
If at a venue with lots of warm lights, get a fan or place the laptop between the drummer and the fan. We also used one of those little USB fan things, pointed from the side onto the laptop's keyboard. This usually helped a little.
 
 
Or you could invest in a super powerfull bad ass i7 fitted mega laptop with billions of megaherz and bytes! But that is probably more expensive than getting an interface with two stereo outs and an SSD

I'd also check cables...loose/dodgy cables are an oft-missed cause of random problems!

Update your interface drivers and firmware.
Perform a defrag of your hard drive.
Have a look at latencymon.

Unless there are really really bad vibrations which may mislead a member of the audience that they are having a minor earthquake I'm not convinced there is something physical going on...

Yup check the cables..

>  I realize that one interface is best, but the fact that it plays fine when the band is not playing live should mean potential conflicts are not causing the dropouts, right? 
 
Not really... it could mean dodgy drivers (not handling record very well), fragmented drive (more read/writes), a process screwing up your latency that is fine during playback.
 
Recording may use different resources/more CPU power.
Oh and BTW if there were any underlying hardware fault (temporary or otherwise) I would be looking at the windows event viewer for it. Generally an underlying hardware fault would crash your entire OS (unless it's cables or something external plugging into it).

Another option to consider is if this is only one track to put it in something that runs off an SD card (like cellphone/tablet). Most take a generic stereo 1/8" out and can run the audio track with a generic player with the right cable (stereo 1/8" split to 2 mono 1/4"), but that cable would be unshielded (possible grief caused here).