The ProPest Silverfish Monitor and Trap contains no insecticides, only a physical trap and a food attractant, so it should pose no risk to people or pets.
What you are describing is not consistent with silverfish behavior. These insects gather in moist areas where is cardboard, paper, or similarly textured items like in or around wallpaper, stored papers, and bathrooms and attics. You may want to use some glue boards in the areas where you suspect the insect to help you catch and identify any insect pests in the area. If you are not able to identify a specific insect pest, then you may consider visiting a dermatologist and explaining the issue and they may conduct a skin scrape to look for parasites or run other tests to help find the nature of the issue.
You may also consider using clear packing tape and lightly pressing it on your skin when you feel a bite to get a sample of the insect. You can then have tape inspected by an entomologist in your region (you can usually find entomologists at local universities).
We wish you the best of luck in getting an insect identification and in finding relief soon!