I made the jump from 135 to SWA in the most recent round of hiring, so it can be done. Actually, my class had at least 4 guys with business aviation backgrounds.

I highly recommend that you take on any additional responsibilities that you can, such as instructor, check airman, chief pilot, or D.O. I held all of those positions in my 5 years at that company and I'm sure that helped a lot. Also, fly as much as you can, especially those TPIC trips; it's hard to build time in most 135 operations, so don't pass up the chance because your 121 competition racks up flight time like it's water. And network your butt off. Lots of airline pilots still own GA airplanes, so you never know who's sitting next to you at the FBO.

Stay focused on your goal and don't get too comfortable at any of the intermediate steps. Keep moving forward. Consider getting some 121 experience to round out your resume (I spent some time at ASA but never upgraded), but not before you get at least 1000-1500 TPIC. It's worth waiting for that, because it will prevent certain doors from immediately being closed.

Good luck.