The client has sent me barcode in 72 dpi (233x82 px, JPG file)

The package design project in Illustrator is 300 dpi. I don't know if I place 72dpi barcode in it, would it print well enough to scan well. On the other side, if I make it 300dpi would it be big enough to be functional.

And related question: Is there a simple way for printer to test barcode when make test print? Is it usual to ask printer fr such thing, thanks!