If you go to Menu and select Edit All Properties (I think it is) you will see 2nd column Name which is by default duplicate Product name. This column is used to display on menu buttons, not product name actually. Let say Product Name: burger and Name you change to My Burger. The menu button will display My Burger but in ticket will still be burger.
if I change the price I would have to do it in 2 places, that can cause errors
Yes, you are right. Price is depend on portion as well. I don’t know if you use portion. Why would server need to see the price? Just add item if you want to check the price. It should not happen often.
btw, there is SQL script from @markjw somewhere on the forum.
There is an SQL script you can run to do it automatically, read more here
beat me to it 
Yeah I did the script for the custom product tag and @QMcKay updated it for doing price
excellent, works well