Are you burning M-Disc DVD or M-Disc Blu-ray?
According to M-Disc, most DVD drives can read their M-Disc DVD's. You shouldn't have to change the drive unless you've already tried this and had the laptop drive fail already. Have you finalized the disc after burning? Many software burning programs, like ImgBurn, will do this automatically.
M-Disc Blu-ray conform to Blu-ray standards and are readable by any drive that reads Blu-ray format.
Instead of replacing the existing drive, you might want to consider an external drive, using USB to connect to the laptop. It is possible to replace laptop drives, but I'm not the one to advise you on that. Maybe Albert or one of the others drive experts around here will show up and help.