I don't think lying is wort the risk to be honest.
Either your MH problems are no problem with working with children ar tey are.
If they are not related and will not be any risk in working with children, you will not be stopped. I was totally cleared to work with vulnerable adults and newborn babies with uni and the hospital being fully aware of my MH issues. All that disclosing did was open me up to further support including counselling and readjusting of hours.
You are right that if you lie, and are caught, that is your career over. Can you guarentee that you will now be totally stable forever more and not need any support at all? Because if you have further MH problems in the future, that fact that you have lied will be apparent.
It is possible to work in these areas. You can be a worthwhile, safe and enjoy a career with MH problems. Just depends on what they are, and I would think, the fact that you have been offered a job and managed to complete your training shows a whole load of strength and ability. Don't risk ruining your chances now. Teachers with MH issues, no problem (when my daughter was in yr1, her teacher had evident MH/depression issues, took time off, but also one of the best teachers she had, encoraging her to be an excellent reader and giving her masses of confidence). A lying teacher simply isn't as acceptable in my opinion.