Would they go out seeking small exposure opportunities like that? Doubtful. Would they readily take an opportunity to slap their logo somewhere for exposure if someone offers it to them and it doesn't cost them anything? I'd be willing to bet yes. Just because they are big doesn't mean they will brush off even small exposure opportunities that are handed to them. Just because they are big doesn't mean they will only enter into multimillion $ deals that require significant amounts of attorney time to construct a complicated agreement.
Would they put a VP of marketing on the job for something like that? No - they probably have policies in place with one or two people to process trivial little things like this as a matter of everyday business. Why would I say this? Because I used to work for a large corporation and was involved with just such a process.