- add a product parameter to the function prototype

This would be too invasive.  Are you using the telit gprs-context driver?  If so, you'd be better off changing that one to accept a custom data structure instead of a GAtChat.  See drivers/hfpmodem for an example.  Or write a custom driver for your needs...

- change the vendor parameter with a structure of two variables

See abov

​This is probably the most suitable solution for me.
Thank you.