Look on the CruiseCritic website to find the sq ft of your stateroom to check of the travel crib will fit or be in the way while the stateroom (or suite) is occupied. I’ve never seen a crib onboard. I know some cruise lines provide travel cribs. I have friend who requested one on a Carnival cruise & the cruise line provided a regular crib (like you’d see in a daycare center), but it was all metal and had wheels on the bottom. Seemed unsafe to me since it had wheels & you wouldn’t want something that wasn’t so top heavy. The closer to the ground, the more safe it would be if the cruise is a little “rocky.” You could also call the cruise line or your travel representative & ask what kind of crib they provide.
As far as a SeaPass card, I don’t think the baby will be issued one, but you will be required to have one. That should be able to get you into the room. Unless you’re in a different stateroom. But you can always find a hospitality crew member on deck. If you inform the room steward, upon the day of your arrival onboard, that you’re the nanny, they should be able to let you into the family’s stateroom should you have any problems.