If they wish to, then the answer is yes to both questions, I'm afraid. I'm not commenting on whether it's "right", "fair", "compassionate" or anything else - those things are subjective and have no relevance to the questions you asked. Do I'm sticking with the questions.
Whether it makes sense or not, the management have laid down a work process. Unless they allow flexibility then that is the process you must follow. You can "advise" then that there may be other / more efficient ways, but you cannot tell them that you can't do it their way. If you refuse to do it their way or say that you can't do it their way, then it may be a performance issue, or even a refusing to follow instructions issue. Either would / could ultimately result in dismissal.
2.5 months off work in a year, regardless of the cause, could result in absence management processes. Those will also ultimately lead to dismissal if absences continue. How quickly that might happen would depend on whether you are already in such processes, what the policies say, etc etc - it depends on specifics for your organisation, but ultimately they are all designed to terminate the employment of people who take too much time off work.
Neither of these things are likely to happen very quickly (unless you refuse to work in the way you are told to) but at best that simply means you have a bit of time. You need to address finding another job before the inevitable happens. And I'll also point out that a lot of absence from work can adversely impact on your chances of getting other work, so you are correct that being off sick isn't a solution to your problem, and could just make it worse. I'd advise getting another job as quickly as possible.