Badlad said:
You say you have been asked "indirectly" by the landowner to remove all your media for this particular cave. This appears to be quite an unusual request if not unprecedented in the area.
My advice would be to arrange to meet the landowner in person and listen directly to his request. Discuss this openly with him and find out the reasons behind his request. If they seem reasonable then do as he asks, or make up your own mind based on first hand knowledge at least.
You had permission to be in the cave in the first place. You acted reasonably in taking the photos/videos and placed them on social media or whatever, as anyone else would. You're not commercial or fallen foul of any rules. Speak direct and make sure you are getting the full story and decide accordingly.
If it is in the Dales you could try the access officer of the CNCC for advice. PM me if you like.
Every situation is different and has to be taken on its own merits.
Years ago, when I was 19 years old, I asked the management for permission to film (Super 8 film) on the Palace Pier in Brighton and had the full TV/Commercials/Feature Film rate thrown at me. Gulp! So I haggled and managed to have the charge reduced until it was a manageable amount, if anything.
In another situation, many years later, I wanted to film inside Wookey Hole Cave (by the Witch, in Chamber 3 and outside the resurgence) for my film on the history of diving and had no problem at all when the cameraman and subject made the approach to the management: Rob Parker and Rob Palmer.
Within reason, we were allowed to film as we wished, scooping the tripod and two cameras out of the way as every new batch of tourists made its appearance inside the cave, at what seemed like five-minute intervals. We also had lights, with cables attached, mounted on tripods, to keep away from stray feet as well!
So, the filming proved episodic to say the least, but we got what we needed and, to this day, Wookey Hole Caves has a free advertisement every time someone watches the end of part 2 and/or the end credits of both parts of the video-film!
On another occasion an archaeologist working for the Tower of London wanted to film a piece to camera with me in the moat beneath the curtain walls - involving him, myself and a Hi-8 video camera on a tripod, on a wide expanse of grass, away from the general public. It was to demonstrate and explain some recent archaeological finds from the moat, with which he was involved.
However, the Tower of London 'PR' department miserably
refused permission (on insurance grounds, I believe). This was after I had time-lapse filmed from Tower Bridge the River Thames going through a tide cycle on the foreshore in front of the Tower of London and made the edited sequence freely available to the Tower management for a public exhibition.
Later, the day manager of the associated Thames Archaeological Survey (1998-9)
lazily refused to help put a schools' teaching film project together, with accompanying text, through which the film could have received an educational distribution to schools and associated sponsorship to help pay for the studio editing and any other unplanned costs. Ultimately, I decided enough was enough and shelved 20 hours of footage and, to this day, the film has never been completed or seen as a direct result . . .
So, I think, prior research and discussions, where necessary, are always best for agreeing in advance how you will proceed with specific photographic and/or filming work. Otherwise, it might all end up having to be archived away, if you work on trust and later find out you're dealing with jokers!