It has been a while since this question was raised - "Is anyone familiar with Oil of Oregano helping with MAC?" So I thought I would search Google Scholar again.
That answer is still - they have experimented with oregano essential oil, among others , in combination with MAC antibiotics, in the lab on slides. Yes, the oil did seem to make some of the antibiotics more effective at penetrating the biofilm to lower the infection. The conclusion - more research is needed. ( https://www.mdpi.com/2079-6382/12/9/1409 )
Here is what we still need to know:
Does it work inside the body? If so, how do you deliver the effective parts of the essential oil to the lungs to penetrate the film?
Does is work to kill the bacteria without antibiotics?

So today's answer is - promising, but not ready for prime time. We lack any evidence that consuming oregano oil or any other essential, will allow needed compounds to reach our lungs and heal the infection. There are strong sensitizing compounds in oregano and other essential oils, so please do not inhale or nebulize them.

One thing we do know that can reduce mycobacteria reproduction is 7% saline solution inhaled via a nebulizer followed by airway clearance to expel mucus. Do you currently do this?