A good approach to take is called "shaping". This is where you train in tiny steps. Once a step is mastered, you move to the next step.
Begin by describing in concrete measureable terms what your current situation is with your cat and doing dental care and your goal for your cat. This is the what, where, when, how long, how often, what tools, what treats, what people will be involved, analysis.
Then, think of many small steps that are in between these two points and are closer and closer approximations of the goal.
For example, having the cat become comfortable with you just petting over the muzzle area might be a first step. (comfortable ... as in doesn't rend and tear the skin from your body for having the audacity to touch there!) If the cat is very mouth shy, you may need to start with petting on the back of the head and gradually moving towards the mouth.
Identify what works as a reward for your cat - treats motivate some (low carb, please!), toys, praise, or laser pointer tag may motivate others.
Now, conduct 2 or more 5 minute training sessions spaced out across the day. The more sessions you can train a day, the faster you are likely to make progress.
You do the behavior, such as petting the head and sliding toward the muzzle. If the cat tolerates it, you reward. You do this for 5 minutes, then stop. Repeat this session at least once more during the day; several more if you can squeeze it in. Continue this for several days.
Once the cat is completely comfortable with a step, you progress to the next tiny step - perhaps that might be rubbing the lip and slightly lifting it. Again, if the cat tolerates it, you reward. You do this for 5 minutes, then stop. Continue this for several days.
If the cat balks, back up to the last successful step (kind of like the last successful dose) and consider if there might be another, smaller or different step which would allow progression towards the goal.
Above all, have patience. This is not likely to be fast for a cat who have never had a human poking something in its mouth!
Note that if the mouth is sore anywhere, you may accidentally touch that area and get swiped or bitten, so wearing long sleeves and pants is recommended. You may need to back up a step if that occurs because you will have associated pain with the behavior. I don't know if there are any topical oral anesthetic products that can be used; if so, that may help in the case of known infection and likely sensitivity.