So the cyst has surface access as you say you can burst it? Has it not occurred to you or your dentist to attempt to clear the infection through that route?
Try soaking for ~10 mins a night in salt water as you would with wisdom teeth. And alternatively with other antibacterial regimens.
I'm going to say a strong no implant! Dental implants are not permanent- look up their life expectency. Often they cannot be replaced after their end of life due to bone loss. Further there are many more complications and costs to you which are trivialized or not addressed. Such as many sick days from work, months/years of pain, trouble eating, nerve injury, sinus perforation, improper placement, and even titanium allergy. Look up 'dental implant malpractice' for some horror stories and a reality check. Don't blindly trust what your dentist sells to you as I did.
By pulling the tooth, if necessary, the teeth behind it naturally move forward to fill the gap. Replacing the tooth is not necessary.