If there's an alternative solution that doesn't cost money, great. Cost aside it is a logical position, following what she's told us.
Flippant suggestion: buy a dictaphone. All arrangements made in person, recorded and played back as required.
Stated more generally, take all of my previous post up to the penultimate sentence and alter to '...such, you will insist upon a mutually practicable and reliable method of your choosing and this will be the only method used.' That could be diaries, letters or email, or conversation followed up in writing.
It worries me that the OP thinks that adopting this, logical, reasonable position, having given him every opportunity to make his preferred approach work, would be petulant. It would not, it would be in her dd's best interests.
The ex is a tough nut to crack because he is determined to be unreasonable and happy to upset his dd in the process. Hence 'no written record, so i can piss about and mess with dds impression of you as much as i like'.
The OP has her dd's best interests at heart, so is compelled, however unfair this is on her, to be the grown up in this situation, never mind unpopular that makes her with the childish, manipulative ex.