Is the use of betadine tetracycline a good option to close capsules after breast implant removal?

Removing the breast capsule (capsulectomy) is not always necessary during breast implant removal. A capsulectomy can create its own set of problems after surgery: fluid buildup, hematoma, contour deformities of the breast, etc... Also, the placement of drains after breast implant removal is also not routine. So, I have no problem with your surgeon's approach. Seroma (fluid buildup), is however, a risk of breast implant removal surgery. The fluid builds up in the space that was occupied by the implants.
Once you have a seroma after breast implant removal, there are several options to treat it. It appears that you have tried the most common way to treat the problem- needle aspiration. If this hasn't worked after several attempts, there are other options. Tetracycline injections have been used to treat seromas, especially after tummy tucks. I don't think it is as commonly used in seromas after breast implant removal. The most common treatment is to surgically remove the capsule and place drains in order to decrease the likelihood of recurrent seroma.
I do agree, that your surgeon should consider testing the fluid before your followup procedure, especially if you had a textured implant. There is a condition named BIA-ALCL that can present as a seroma formation after breast augmentation. Uncomplicated forms of the condition should be treated with total removal of the capsule. So, consider having the fluid tested before tetracycline injection.
Have the discussion with your surgeon to be on the safe side.
Good luck. 

Hello! It is likely the drains will take care of the fluid buildup issue. If the fluid persists with the drains in, then tetracycline or doxycycline installation may work. This is a classic treatment for persistent fluid buildup. If the fluid seems unusually persistent it may be worth having the capsules surgically removed. This is not typically done for routine cases.

While these things are sometimes used to treat seromas chances are the capsules need to be removed.  I would discuss with your surgeon who knows your case best the chances he or she estimates for success with drains and if you do not like the answers you are getting go seek another in person consultation.  Bring your records with you so the consulting surgeon can have all of the info.  Best of luck!