Whenever I play back my songs on the home stereo, mp3 player, or car speakers, they sound very bass-heavy, with not much treble.  This is in contrast to when I am mixing them (on headphones), in which case the mix sounds more balanced.
 
So, apparently my mixing headphones are biased towards the high end (treble).
 
I'd therefore like find a way to compensate for this, by somehow making the headphones sound more bass-heavy (and less trebly) during mix, so that they more accurately represent what we'll be hearing on the home stereo, mp3 player, and car speakers.
 
Is there a way in Sonar (X3) to apply a bass-heavy EQ to the monitoring output (to the headphones) without interfering with the actual mix output?

I put an EQ on my master bus, with a preset called "Room Tuning" (in your case it would be "headphone tuning" and I remember to turn it off before I bounce a final mix to WAV file.
 
 

I think what I'd do is put an EQ on an additional buss after the Master, and when you bounce the final mix out just do it from the first Master, which would cut out any FX after that point in the chain. Or just put it last on the master and bypass when you're done.
 
Really though, I think you're better off just compensating for it mentally. It'll take so much tweaking/checking to get your EQ settings to really match the system you're shooting for, that you might as well just learn the headphones inside and out. 
 
I hear you for sure. Always an issue for everyone, I think. I have started actually importing reference tracks into my sessions for easily A/B-ing while finalizing/mastering.

How about purchasing bass heavy headphones? I think would be a good way to make you achieve more balanced mixes.
You could also put a low shelf eq to master bus during mixing and remove it just before exporting the final mix.
 
During mixing, listen reference songs and try to make your mix sound similar.
 
Mix by meters. Decide maximum allowed bass volume you may not exceed. Force yourself to value mid and high frequencies, too. :)
 
 

Go to sonarworks and pick up their headphone calibration system.  They have modeled a long list of headphones and their system works wonderfully on my KRK 8400's... you have many options, as you'll see once you arrive at their site.

Vastman
Go to sonarworks and pick up their headphone calibration system.  They have modeled a long list of headphones and their system works wonderfully on my KRK 8400's... you have many options, as you'll see once you arrive at their site.

I agree with all that. I also use those KRKs as well.

Use Beats headphones. Then you'll wonder why your mixes have so much treble 

My RME comes with a factory cheap headphones compensation EQ preset.

I think you should find the flattest monitors you can to get a good mix. As good as headphones are they will give you a skewed  stereo image and can have frequency bumps or hollows that color the sound.  Good near monitors are the best way to approach mixing. 

I keep meaning to pick up that Sonarworks system for my ATH-M50's. I couldn't do without ARC2 on my monitors so why am I denying myself the possibility of flat cans as well? It would also make my VRM box way more useful. 
 
ARC2 is like putting contact lenses on my monitors, the effect is incredible. I have all of my system audio wired through it via Virtual Audio Cable and Pedalboard2. Spotify goes through it, YouTube, Netflix, everything. I'd love to be able to slap Sonarworks on for my headphones as well.