Decent advise but you missed one point
If HR refuse to provide it then write one your self and take it to your line manager and get them to print it on company paper and sign it.
Keep all these letters in your own file for future interviews if necessary.
No line manager will sign any document without HR and their legal dept guidance in a real company so you might not get that. Better to be honest and present any items confirming you to have been there for those dates as proof......