The possibility of silverfish being able to bite humans is an interesting topic. Most pest control experts say that the insects don’t bite people and that any claims that they do are due to mistaken identity. They say that the mouthparts of the insect are too small to hurt humans. They also say that bites, when silverfish are around, are probably due to bedbugs, fleas, or mosquitoes and that a biting “silverfish” seen on the skin is probably a centipede.
On the other hand, I’ve seen anecdotal reports from people who say that they are certain that the insect on their skin was a silverfish and that it bit them before they brushed it off.
I have no experience with insect bites in the mouth. It sounds like a very unpleasant experience.