You need to determine for yourself whether the type of extermination services the tenant is requesting is actually necessary. If the exterminator you hire concurs that there is a significant termite issue that needs to be addressed with fumigation, then I would recommend that you do this and agree to either pay for the tenant to stay at a reasonably priced hotel/motel for 2 - 3 days or offer her an offset on the rent for the following month. This is a matter that should be relatively easily resolved through discussions. You may need to make arrangements for the appropriate storage of any of the tenant's personal property that will need to be removed from the unit during the exterminating process.
Good luck with this.
This is not a substitute for a consultation with a housing law attorney.
A tenant is entitled to a rental unit free from vermin. Since the fumigation will interfere with the habitability of the rental unit for a few days, the tenant is probably entitled to some pro-rata reduction in the monthly rent and/or reasonably comparable alternative living situation such as a reasonably priced hotel plus compensation for the additional food expense.