I think as mediums, psychics et al are all still to prove their real abilities, that inviting a psychic along to an investigation might be useful if it was the psychic being tested - maybe a previously investigated site so some useful base line was set, along with a few other individuals - some not in on the test who could be influenced by the psychic, and a few in on the test to keep an eye on the goings on and see if the psychic can be influenced too. Plus some official investigators to lead the event and set some tasks - readings in rooms etc
No doubt the medium with a few accolytes in tow will be willing to put on a show rather than come to the conclusion that - nope, the house isn't haunted - but it'd be interesting to see what effect the medium might have on the other guests - after all nothing ever happened on Britains Most haunted, but plenty of people imagine more happened than some people running around shouting in the dark for a few hours.
I think if I was psychic I'd love to test my abilities rather than become a novelty act - but it's easy to fool some of the people all of the time and make a tidy living out of it...