It was possible to lower the information that could be learned by that person to make less information be passed on if that did happen, but there was a limit to how much could be done about it.
There was still a limit to how long Ash could maintain her transformations in her present state, and even if she provided a false name and other false information, there would still be other things that could be learned through each interaction.
Even if she did not mean to reveal something, it was easy for her to mention something during a conversation and slip up at any time, resulting in there being a chance that it could be used against her one day.
There were benefits to staying close and gaining more control over the situation, but the risks involved were just too high, and it required a lot more time to be devoted to it.
