Why is it you think all the other professors there think you're incompetent? The three of them that know your work best, besides your advisor, all thought you earned a PhD; one of them hired you to work with them as a post doc. If your PhD advisor is a malicious person, it's quite likely this hasn't escaped everyone else's observation, and they may already know to ignore them. At least in my world, it's incredibly unusual for someone to earn a PhD without their advisor's blessing, and it would be quite a statement about what other professors thought of them if they were willing to override their decision. Professional politics and courtesy may prevent them from stating this more explicitly than they already have.

Anyways, I don't think there's really a right or wrong answer here about how much to push back. You could take the "high road" and let the people who know you best speak for you when you need it: when you're asking for recommendation letters, for example.

If you do feel it is necessary to escalate, your post doc advisor is probably the person best positioned to help advise you on how to proceed. Since you're still at the same institution, if you choose to escalate I would recommend doing so within the reporting structure of your institution. That may involve going to the department chair, or it may involve a separate third-party review process. At some point, you may be in a room with the chair, your former advisor, and your current advisor, and there will be an expectation of some adult professional conversation among you, which your former advisor may or may not be capable of.

Try to stick to what is meaningful to you going forward, not payback or revenge against your advisor. It's quite reasonable that you value your professional image, and are proud of your integrity and the integrity of your school. It reflects poorly on the whole department to have unfounded accusations that graduates have not actually performed the research that earned them a degree. Your advisor looks especially foolish to make this accusation about their own student, since monitoring your progress is their job. It would be as if the chef were whispering that the food they serve is expired. They are also implicitly questioning the professionalism of the other professors on your committee.

Likely a process like this will have the most value if there is some specific behavior you would like to stop; you won't be able to order your former advisor to change their opinion of you as a researcher.

Why is it you think all the other professors there think you're incompetent? The three of them that know your work best, besides your advisor, all thought you earned a PhD; one of them hired you to work with them as a post doc. If your PhD advisor is a malicious person, it's quite likely this hasn't escaped everyone else's observation, and they may already know to ignore her. At least in my world, it's incredibly unusual for someone to earn a PhD without their advisor's blessing, and it would be quite a statement about what other professors thought of her if they were willing to override their decision. Professional politics and courtesy may prevent them from stating this more explicitly than they already have.

Anyways, I don't think there's really a right or wrong answer here about how much to push back. You could take the "high road" and let the people who know you best speak for you when you need it: when you're asking for recommendation letters, for example.

If you do feel it is necessary to escalate, your post doc advisor is probably the person best positioned to help advise you on how to proceed. Since you're still at the same institution, if you choose to escalate I would recommend doing so within the reporting structure of your institution. That may involve going to the department chair, or it may involve a separate third-party review process. At some point, you may be in a room with the chair, your former advisor, and your current advisor, and there will be an expectation of some adult professional conversation among you, which your former advisor may or may not be capable of.

Try to stick to what is meaningful to you going forward, not payback or revenge against your advisor. It's quite reasonable that you value your professional image, and are proud of your integrity and the integrity of your school. It reflects poorly on the whole department to have unfounded accusations that graduates have not actually performed the research that earned them a degree. Your advisor looks especially foolish to make this accusation about her own student, since monitoring your progress is her job. It would be as if the chef were whispering that the food they serve is expired. They are also implicitly questioning the professionalism of the other professors on your committee.

Likely a process like this will have the most value if there is some specific behavior you would like to stop; you won't be able to order your former advisor to change her opinion of you as a researcher.