You can try flux-core lead-free solder. It's typically mostly tin with a bit of copper and sometimes other metals. It's not as shiny as electronic grade eutectic solder (dull grey). The flux is in the core and is sufficient for soldering clean bright copper surfaces together.
I have no information on the biocompatibility of (for example) Sn99.3Cu0.7 solder for long-term contact. You should clean the flux off after soldering with an appropriate solvent such as IPA or lacquer thinner).
Melting point is higher than normal Sn60Pb40 or Sn63Pb37 solders- by perhaps 50°C so set your soldering iron to a higher temperature if it is adjustable.
Jewellery making suppliers may have other options for you such as colored solders that match metal colors, but I think they are mostly silver solders that will require a small brazing torch rather than a soldering iron. Some silver solders may still contain cadmium so take care if you start looking at unconventional sources. NSF approval is a good indicator it's probably not overtly toxic.