Personally I am not a fan of insecticides (I'm a beekeeper), but if you are already in an area where they are being used then it may benefit your situation. My own feeling is that there is a balance in nature and when things get out of balance, as is suggested by your tapeworm issue, using something which creates further imbalance (like an insecticide) may well cause other problems down the line.
In my opinion there are no quick fixes in these situations that don't have potential longer term consequences which may be unforeseen at this time, but you have to make a decision based on your own ideals and circumstances. I'm sorry, I can't tell you what will definitely work and what will not.
I have just done some more research and it appears that it needs to be quick lime rather than garden lime but the birds would need to be removed during treatment. The link below was written by a parasitologist from Darwin, so may be helpful in your situation....
https://nt.gov.au/__data/assets/pdf_file/0004/233779/internal-parasites-worms.pdf