Lead poisoning from a bullet?

I know that it is common practice for a surgeon to
leave a bullet inside of a victim if the act of removing it could cause them more harm than good.

Assuming that the bullet is made of lead,
(and most still are)
wouldn’t this cause the person to contract lead poisoning?

Or is this procedure only followed for bullets not containing lead?

Yes.

Managing Elevated Blood Lead Levels Among Young Children: Recommendations from the Advisory Committee on Childhood Lead Poisoning Prevention

Better info:

Lead arthropathy and systemic lead poisoning

“I hear Ol’ Tex got himself killed out in Dodge City.”

“Yep. He died of acute lead poisonin’.”