Steve's idea will likely not fly as the end buyer likely has a lender and lenders will not usually loan without a clean termite report.
You have just learned another valuable and costly lesson taht $95 could have saved you. Get a termite inspection during your purchase process so you know what issues are there and can handle them during your rehab.
Since you are, where you are and the problem is only in one spot (as indictaed by your post) use plastic to seal off the kitchen, chip up the tiles, replace the bad wood beams in teh subfloor, and replace the tiles. From your description, I can assume that this is an elevated foundation. The other option may be to work from underneath the home (although very difficult).
As to your contract with teh home stager, you have two options:
1. Ask them and get permission, describing the small amount of work that will be done and tha erea will be sealed off during the work performed.
2. Seal of teh area, get it done, and don't tell the company and you will be liable for any damage done to their property. If you are willing to take on that liability, then you may go that route, otherwise, you will need to get permission or have them romove their property, do the work, then pay to have them bring it back.