This appears to have cropped up a few times recently. This is my idea, slightly embellished, from another thread. I don't claim any great originality since I'm sure such ideas have been thrown around before.
Surely this is something that could be at least partially addressed by adding a press conference/statement after a manager leaves a club (or adding questions to the conference after taking a new job). Probably shouldn't apply if a manager has been sacked (you generally don't hear from sacked managers formally until they start doing specific media interviews irl), but if you move on without being sacked perhaps you should get the opportunity to give your reasons and your feelings - Offer I couldn't refuse/Felt it was time for a new challenge/Taken the club as far as I could/Difference of opinion with the board. With the feelings about the move it could be altered depending on your time with the club or reputation there so a long-serving manager who had brought success would be able to say they would always have a soft spot for the club or hoped the fans would understand the move while a manager who jumps ship after a year would be limited to more selfish options (ie, 'I had a good spell here, even if it was short'/'Football is a business and this club was just another step on the ladder for me.').
My concern would be that it might be a bit tricky to provide negative options, (or would be too easy to win credit by lying) but if you're just trying to mitigate an inevitable loss (say 100 points of overall reputation with the club up to the maximum 100 dislike) then it would still work (ie giving the best answers would see you lose nothing - it wouldn't be realistic to gain rep by leaving - while giving some good and some indifferent answers would lead to a loss of 30 points and take you from legendary to favoured personnel or from indifferent to mildly disliked. Obviously saying the club were just another job to you and you were only looking out for number 1 would see you lose the full 100. Only the most loyal and successful managers would get the questions to keep their club rep intact, everyone else would see a guaranteed drop.
In brief, I see it as going something like this:
Manager gets the sack - Nothing changes.
Manager accepts another offer - Release statement based on 3 or 4 set questions/answer extra questions in press conference at new club explaining reasons for leaving and their feelings about their old club and about the move itself.
By leaving the club the manager opens himself/herself to a maximum 100 point reduction in rating (a true legend risks dropping to indifference or mild dislike, a regular manager risks extreme dislike) but can win back points by answering questions. The amount of points they can win back depends on their answers, but also hinges on longevity and success with the club which open better options to the manager.