Nah: you'd go to the loo and grab some loo roll you can flush away and then wash your hands. Or near the gun cupboard downstairs was the kitchen sink and a shower room: there's no reason not to simply run the moderator under the tap and dry it on a dishcloth. or as he cycled back past a big fish pond thing he could have just chucked the moderator and the kitchen phone into it. A tampon would have been way too big to clean clean the moderator and you can't break 'a bit' off one, it would just disintegrate.
This crime has a lot of sexism in it: the tampon theory was a silly attempt to explain why there were tampons out in public rather than wish Sheila's other toiletries on the bed. I find the fact the tampons were out and her stained clothing was soaking in buckets rather publicly in the kitchen, points to Sheila not being herself as they were surely items to be kept out of sight of her father, sons, brother and people who worked at the farm early the next morning making themselves tea.
These murders we planned, whoever did them: Bamber had hours to sort himself out if he wanted.
I wonder if the murderer was hiding in the sewing room to ambush Nevill as he came upstairs after hear June be shot. Or perhaps if it was Sheila in the grip of psychosis she barricaded herself into the small room to escape whatever demons she imagined were outside it. In a previous episode she had told a friend that God was sitting in the room with them, remember.