It looks like there is something wrong with your eyes, and eye bags are really bad and make you less confident. From what I know, toothpaste is quite an irritative product, which might give rise to some allergic reactions on your eyes, making your eye bags even more serious. What I recommend is some good cosmetics, good way of life, nutrition and vitamins, and most importantly, a good mood. Hope you find this useful.
Toothpaste does have a lot of worth more than cleaningour mouths, such as reducing inflammation, stopping bleeding, relieving internal heat or fever, ease pain, remove blood stasis and deodorizing, etc.Bags under eyes appear when the skin lack of elasticity or lipomatosis.
You'd better not to use toothpaste to the skin around the eyes because the skin around eyes is too thin, and it won't help you. You can try put walnut oil or tea tree oil in your daily meal, they would help you.
I don't suggest you to use the toothpaste to help the bags under the eyes although it has the role to reduce it. However, the toothpaste will stimulate your eyes skins. I suggest you to do the warm compress to make the eyes feel comfortable and reduce the bags under the eyes.