Since you still have strong feelings for her, and she's clearly not interested in getting back together, I think it would be really difficult to be living with her for a full month. If I were you, I would consider staying somewhere, or with someone else. You just mentioned that you will still see her when you're in town, so there's still a chance that you could rekindle something in that time. However, it sounds like she's pretty clear on the fact that she doesn't want to travel away from her family for long periods of time, and she's probably not going to change her tune with you leaving again in 4 weeks.
Obviously if traveling is part of your career, and you like what you're doing then you need someone who is willing to travel with you, or who is willing to do a long distance relationship. It doesn't sound like your ex is that person, so I think you should avoid her as much as possible to spare your feelings, and focus more on finding someone who is comfortable with your lifestyle.
If your feelings for her are more important to you than your job, then maybe addressing your feelings with her would be a good idea. Of course, in order to make things work with this person in particular you might need to find a new job, one that doesn't require you to travel and allows you to be with her.
At this point though, she might not have the same feelings anymore.
Consider what you can't live without, her? or your job? if it's her, then you need to let her know how you feel and do whatever it takes to be with her.
Good luck!
