Hi @Chuckdog
A very warm welcome to Sane. Sorry to read that your partner lost his job due to wrongful accusations - being unemployed, especially under those circumstances does tend to trigger depression in most. So I can understand your partner's struggle for the last few months.
Its good news that your partner has another job. Well done on his part! If your partner was not put off for poor work performance, I can't see why he can't hold down this job? Why do you fear he will struggle - due to mental health issues?. Is he under the care of medical professionals whom are helping him to keep his illness under control? It really depends upon the suitability and stress levels of the work he has undertaken and how well controlled his mental illness is - if the work is what he is use to and he coped with that - he should be fine.
I think what would help him is to build up his confidence at this point. Even though you have your doubts and anxieties try and hide this from your partner and encourage him by saying he can do this, and by doing his very best is enough. Be proud of him for not giving up and finding another job. That takes drive - let him know you are proud of this achievements. He may just surprise you. Planting the seeds of doubt now would only feed his own anxieties and mental symptoms giving a bad start.... If he goes into this job with a positive attitude this will give him the best chance of success.