I have been on holidays and had arguments, if you are not getting along then doing seperate activities is perfectly normal during trips. In your case, I think this is especially true because:
she made it clear to me that she doesn't want to see me outside of work any longer
Firstly, this calls into question whether your intuition that "I don't think she will be okay with having to go do activities on her own" is correct or not because it is the exact opposite of what she has attempted to make clear previously. Even holidays where I am getting along great with everyone can include moments where I need/want to do something alone and this is fine. Of course you want to help each other out, be there when needed and try to have a good time together but that does not mean you are responsible for keeping the other entertained 24/7. Trying to do so may end up wearing you out, suffocating her and creating a worse time for everyone.
So, considering these things I believe that being polite and avoiding animosity invite a different goal than explaining why you don't want her to go with you. Instead, I think politeness will be more about making sure your absence does not suddenly cause inconvenience.
This means telling her early enough so that she can make new plans. She may not always tell you her plans, she may have already paid for something, she may have been hyping herself up for something so it is important to let her know as early as possible. It also means to ask and being willing to compromise about the usage of any shared resources. If you have one car, or even something that seems insignificant, be sure to check if its okay for you to use it.
How you inform her is up to you. I think the other details mentioned are more important for staying polite but as a general template @Robin's answer is more than fine:
on 'Date' I am planning to meet up with a long time friend from high school and we have some catching up to do. It will take the better part of the day, so you might want to look for some activities you can do on your own. - @Robin
Some general pointers for limiting an aggressive escalation of conflict is to remain calm, and when giving reasons try to focus on yourself. Putting the focus on the other person in situations like this leads to defensive retaliation and conflict escalation. For a quick example, saying "I want to go alone" might be a good alternative to "I don't want her/you to come".
Edit in response to comment:
The fact that she is glad to not be your personal friend, and that she bluntly stated that she was prepared to endure your company only makes me feel stronger about my answer.
All of her behaviour indicates to me that this will be different from your previous trips. She has made it abundantly clear that she is trying to distance herself from you and yet you are still planning the whole trip with the thought that spending part of one day apart is a big/bad issue. She is not a child, and acting like it is your job to make sure that you two are doing things together the entire time after she has told you she does not want to spend time with you will only smother her and raise tensions even higher. Even if you are right, and she really does expect you to spend every waking minute with her that is an unreasonable request and it is not your job to adhere to it. You have a right to do something for yourself while on holiday and there is nothing wrong with letting her know and calmly/responsibly informing her of your plans and if her actions are at all a representation of how she feels then she will probably appreciate the time alone anyway. You may not understand her reasoning, you may wish to be closer again but taking on this role is not helping either of you. Your only actual responsibility is to... be responsible. Don't worry about how she will react so much that you put off telling her until the last minute and ruin some plan she might have had. If anything, that is what might create animosity. So be polite, organise the details early, meet up with your other friend and as is very normal even between good friends, give her some space... after all, she IS asking for it.
