You cannot answer 'no' - as you say yourself, it is obvious that you intend to stay longer term. You therefore have to answer 'yes' and supply the medicals.
The visa(s) are the priority here, NOT the job. You HAVE to fulfil the visa requirements (which include telling the whole truth to INZ), even if they're inconvenient, and even if your employer turns out to have to wait past your wished-for start date. In the event, though, the work visas may come through in time for 22/10. If they don't, you certainly wouldn't be the first new employee that ADHB had had to wait for because of issues with INZ. I think you'd find they'd see the situation with more calm than you're expecting, as long as it's obvious you've done everything you could and not sat on your hands about making your arrangements.