It will not damage your case unless your new lawyer is not as good as your prior lawyer. Attorneys will have to split their fees at the end, but you need not worry about that part. The fee total is set by the judge at the end and is usually 15%.
Attorneys are not supposed to charge injured workers anything. If a lawyer wants to take over your case and represent you, he or she may feel a writ is warranted and/or justified. It is often hard for attorneys to try to fix what was done by pro per applicants once it reaches this level. The Court of Appeal will only hear the legal issues raised below and nothing new.
There is a difference between "public record" and "discoverable." With very few exceptions, a deposition transcript will probably be discoverable in a related civil or criminal matter. Since you are talking about job loss, I'm assuming there could be a wrongful termination civil case filed. If so, the employment lawyer will probably get a hold of this deposition.
I love the title to your question. I ask myself this question daily. The likely answer is because of the insurance lobby in California.
The most likely answer to your TTD question is that the QME probably said you are currently TTD. You really should not try to handle your case alone without representation. You can have a professional attorney handle your case for usually only 15% of the end award.
Once you file a WC case, it becomes easily searchable and part of the public record. It would not be surprising if an employer did conduct a thorough background check, which would include WC claims check. However, it is illegal for employers to discriminate on the basis of disability. If you get evidence to support your hunch, you could have civil claims under FEHA or the ADA.
I think you need to retain a lawyer ASAP. A lot of these clinics fail to properly diagnose a major problem that may require surgery to fix. A lawyer will get you to either a better PTP or request a QME/AME. You will not lose medical benefits by accepting a new position. Please get a lawyer ASAP.
Everyone's answers above are wise and correct. I will just add the cliche, "the truth will set you free." So long as you do not lie about something, the video and surveillance means nothing but an extra bill for insurance companies to pay.
It is completely unethical for a lawyer to not let his client decide about settlement. It is your case. Your choice. The lawyer can only offer his opinion if it is a good or fair settlement offer.
By the way, you are always free to change attorneys -- especially if there is a breakdown in the relationship.