Carbon-14 is not the type of nuclear waste anyone is worried about. It's barely radioactive, with a long half life and very low decay energy. Disposing of it doesn't pose any issues because it is unable to make anything else radioactive.

You want to make a long lasting low power nuclear battery? Go for it. But don't try to trick people into thinking you are doing anything about nuclear waste.

The nuclear waste that causes problems is the type with neutron decay that is able to activate nearby elements. That is tricky to dispose of, and difficult to store.

You didn't specify whether it was waste anyone worries about. You claimed the material has to be made specially for these batteries. That's not the case, it's literally a radioactive waste material from nuclear reactors.

According to the article I linked, it does need to be safely stored, since it's dangerous to touch or ingest, and it needs that storage for a long period of time.