Is there any advantage or disadvantage to the type of request I make? In other words, will I have a better chance of getting a grant if I apply for sponsorship instead of general operating support?

We treat each type of grant we make equally, so there is no advantage or disadvantage whether you apply for general operating support, a project grant, sponsorship or capital grant. You should choose what you request based on what you need most, and how you can leverage other funders to support your work.