As others say - work permits and visas are different things.
You’ll need an work permit from the language school that hires you. Doesn’t matter if you are married to an Indonesian or not as this is a formal job
The work permit will also allow you to get a residence visa (kitas). You can also get one of these by being legally married to an Indonesian
But a KITAS doesn’t give you the right to work in formal jobs. However, there is an advantage to having your KITAS from your marriage status not your job, as if you get fired/ decide to quit you can still stay in Indonesia
You should be able to survive on a teaching salary and support family. But think motorbike/ scooter rather than car, small rental apartment or living with her family. It is quite odd for wives/ women not to work in Indonesia, even if ‘informal’ work like catering. Tax rates in Indonesia are comparable to the U.K.
One issue - if you are ethnic British most Indonesians will regard you as that nationality, rather than British - I.e. you will be seen as Indian/ Pakistani. This can have pluses and minuses. There’s a long established South Asian community in Jakarta.