NOT A CHANCE. You'll never see the inside of an engine room. I can guarantee that. Nor, the bridge...unless you're on a long cruise (like a crossing) and the captain is really great.
After 9/11, everything changed. People got paranoid. We're more likely to drown in our sink than be the victim of a terrorist attack, but supposedly they're everywhere...like in your pants or under your bra at the airport. Sit me in front of FoxNews for a couple of hours, and I'll start thinking my parents are hiding Osama in their pants...it's unreal how stupid it's gotten, and it's a shame, because those were always awesome tours. Like kids seeing the cockpit of a plane. I got to do that. My kids won't...all because of some guys in a cave with box cutters, that we probably made up anyways. We've forgotten that people are usually decent. It's a shame.
Most crew can't ever go down into the engine spaces. It took me years to be able to go down there, and I only ever went a couple of times...and always working, never just to have a look. That doesn't happen anymore. We make videos and put them on TV to show people those spaces.
Never hurts to ask. Upgrade to the owner's suite, and maybe you'll have a better chance. Buy a couple million of the company's stock...even better.