Having worked for many years as a specialist dental nurses, I've seen this problem many times. What the dental hospital suggested is an operation called an apecectomy, which is the removal of the tip of the root of the offending tooth. This will require a small piece of bone to be removed through an incision in the gum. Quite often an X-ray will prove that there is an area of stubborn infection around the area, this in turn makes it risky to dive in, thus spreading the infection around. I am unsure why the dentist has told you that you could have this done privately at great cost, I do hope therefore that it is not just because if anything goes wrong the responsibility will rest outside the nhs! Iwould strongly recommend you ask for a second opinion from the consultant oral surgeon at the dental hospital.
Don't move until you are granted a consultation and then ask for a full explanation.
Good luck. 