Following up to add that I found your post interesting so I did a search "training company submitted false resume" on google. Holy moly it looks like you have a lot of company! Crazymotts has a particularly interesting blog post on this with good graphics and many comments if you want to look that up.
I had no idea that this was an industry, but apparently this "business model" of having people sign restrictive contracts and then creating false resumes is big business for the middle men! The hammer is going to come down on this sooner or later, and my opinion is that you need to get out of this situation as soon as you can.
Ask yourself: did you sign the contract before or after they falsified your resume? If you signed it at the beginning of the training program with an honest intent, then that works in your favor. If so, then I think that to some extent they trapped you into this. If so, then I think it would be really hard for the training company who did this to successfully come after you if you decided to breach that contract by quitting the contract work and accepting a full time position that immediately cuts them out of the deal. If they send you anything threatening in writing, which they probably will, reply in writing only "I dispute your claims" and write "do not contact me again." You can follow up with "Please direct all future correspondence concerning this matter to my attorney, InsertAttorneyNameHere, JD, insert attorney's address." Keep any written response to the training company or recruiter very short, or even better just have your lawyer respond on your behalf.
You still have the problem of a false resume hanging over your head, of course, so consult a lawyer. I agree with the suggestion that you should make a true resume with the current experience on it to move forward with and forget about the fake one they made. If they are truly this shady, I would completely leave the training company's name off of anything associated with me in the future. You can still have a "skills" section that says "fluent in java web programming" or whatever skills you have without mentioning this shady company. Do not ever put the "fake experience" down on any job application that you sign in the future, and that includes with your current company if you have to fill out their application to convert to full time. In a few years this will all be behind you . . . maybe sooner.
I hope that you do consult a lawyer, who will probably tell you to immediately stop posting about this on the internet. So, we will likely not hear from you again, and I wish you the best.
I am not a lawyer and this post is not legal advice, just my opinion.