I don't know what gravel/road/ rough road experience you have, but my comment is take it easy/ slowly as it will reduce the chance of doing any damage to your vehicle or trailer. We found plenty of "bat outahell" drivers who were a real menace to themselves and others coming the other way.
The "Tip" should be just part of the journey , not the holy grail as some of the drivers seem to behave.
If in doubt about needing to replace any suspension parts, do it before hand. If your shocks/struts and possibly some bushes are getting towards 200 000km or more, replace them before you go.
Allow at least two weeks to get from Cooktown to the Tip and back to enjoy it.
Allow time to go Thursday island for the day.
There is a Cape York tourist website.
For us it's a been there once, not bothering to go back experience, plenty of others go there numerous times though.
We prefer more western locations and southern beaches where you can swim without blasted crocs.
Mark