I'm no native speaker, so I'm waiting to be corrected. I think "I would have loved to have been part of that." refers to a situation that has already passed, so you have to use a tense of the past. It's not possible to be part of it anymore, because it's over. I'm not quite sure why the second "have (been)" is necessary, I would have used "to be" here. "I would love to be part of that" refers to a pesent situation or to one that's coming soon, so it's still possible to be part of it. Hence you use a tense of the present. Did I get that right or is it complete rubbish?

kim, -I would love to be part of that. The present conditional refers to the present or the future. -I would love to have been part of that With the past participle (have been) it is a present regret about a past event. -I would have loved to be part of that. The past conditioanl refers to a past regret about something that happened in the past. -I would have loved to have been part of that. A past regret about something that happened before in the past. ----- On opening night (in the past) I would have loved to have been part of the writing (writing that was done before opening night).