Advice on oral care for a person with dementia?

Though my experience is different -after moms stroke (before she was awake and able and cleared to swallow) we used the oral care sponges just to moisten her mouth and get a clean layer on her teeth. We then used them going forward with speech as she started to wake up.
Someone mentioned using them to check for and clear pockets of food which is a great idea.
Im not sure how difficult it is for you to help your patient but speech also have us a very specific mouth rinse for mom in the early days (it was less harmful if she swallowed some).
Are you more worried with someone who wont allow you to help or are you worried about swallowing and aspiration?
They do make a suction tooth brush (most insurances will not cover it) but I do remember there being another one that was a private purchase). If you are worried about aspiration and swallowing try looking up suction toothbrush. I would also recommend getting some of the oral sponges (your loved one may even be able to use them with your help).

I've done a site search for you, hope these are helpful:

https://www.agingcare.com/articles/oral-health-tips-for-dementia-caregivers-203144.htm

https://www.agingcare.com/articles/oral-care-strategies-older-adults-175335.htm

https://www.agingcare.com/articles/bathing-hygiene-top-tips-assisting-with-dental-care-431727.htm

It can be tricky.
First thing after each meal use a swab, there are ones designed for doing this, and wipe the inside of the mouth along the gum line and the entire cheek area. This should remove any food particles and any pocketed food.
Encourage a drink of water after each bite of food. This will also help flush the mouth after each bite. (even thickened water will have the same effect)
If you can use a toothbrush with a toothpaste that does NOT have fluoride in it. Brush as best you can. An electric toothbrush makes this easier. And it is very possible that the person may "help" you brush their teeth.
If it is impossible to brush the teeth using the swab will help. You can get some that are plain and you can moisten with mouthwash (non alcohol) and there are some that have a product imbedded in it that is "like" a toothpaste.
Under no circumstances should you put your finger in the mouth.
If the person has teeth and they are not in good condition it is a tough call to do dental work or leave it as is. Most dementia patients will not be compliant so the use of anesthesia is necessary and that is not a good thing. If they are compliant and a tooth can be extracted you are then left to monitor the wound until it heals. And keep the person from "fiddling" with it.
As with many things there is a "Benefit VS Burden" when it comes to some choices that we have to make.