If there is an error in your log, just strike it out, and make the correction, everyone makes mistakes, and they are easily forgiven if understood. Things written under whiteout are hard to understand.
You may not get to write some flying you do into your logbook, but that doesn't mean that that flying does not have value. You flew the plane, you learned a whole bunch of new stuff about bush type operation - excellent! You got the photos, and smile on your face. You will be a better pilot for this added experience. So you're better than your logbook indicates you should be, perfect! Hopefully you have lots of opportunity to fly aircraft for which you cannot log the time! It means your flying is diverse. That's better than only knowing the cockpit of one 172, and logging all of that time.
The other thing you can do is to make the entry for flying the Caravan, but not total in the time. There's no harm in recording flying that you have actually done, there's just some flying, for which you are not entitled to take credit for the time. The authorities who read your pilot log are not trying to catch you at anything, they would just like an accurate record of the flying experience you have, for which you are entitled to take credit for. They're also happy to see the experience you have for which you cannot take credit, as long as you aren't taking credit!