Yes, it's a good idea to meet with HR to confirm what is possible and what isn't. Most large organizations have policies that outline what they will / will not provide in these situations. Since your company is a large company, it's safe to assume they have such policies in place. If this is against their policies, then there won't be anything you can do to change their minds. It simply won't be possible for them to complete the required paperwork and you'll have to look into different PR programs.
There is no fee. They need to complete the employer form.
I would start looking into other PR options while you wait to meet with HR. Since they answered you so quickly, it's most likely against their policies to complete the employer form. This is unfortunately quite normal for larger organizations.
Good luck.