If I were you, I’d want to do what is needed to treat syphilis. If left untreated, 10-30 years later it can cause issues in multi organ systems. It can be cured if treated early. Looks like Benzatine is the correct treatment, it is a penicillin. I’m not a medical professional but it seems like a very safe treatment unless you’re allergic to penicillin.

As a mom myself, I’d suggest telling her. My sons have told me some really upsetting things over the years - but life is life and my job is to help them thru it. I don’t know how your mom will respond, there are some people who have horrible relationship breaking responses, but I think most moms need to know. She will be upset, share her grief and upset because I’m sure you’re feeling the same thing. Then get treatment.
This episode, syphilis and treatment, will probably end up in your health history somewhere. And, especially if you’re on her health insurance, it may be noted at some point. Health histories, like those recorded when you start a new doctor or if you get pregnant might capture that info. You’ll just have to make the best decision you can for now - but if it were me, I’d absolutely chose treatment.

Here’s more info on syphyilis and side effects (far down the page).
https://www.ncbi.nlm.nih.gov/books/NBK507723/