If you have to go to the trouble of removing the faucet to remove the spring, you might as well pick up a faucet-thread line cleaning adapter. It's a simple fitting that threads directly onto the shank once the faucet is removed. It has a hose barb so you can attach some vinyl hose, and direct that to your reservoir that you're recirculating from. Then, while recirculating cleaning solution, you can disassemble your faucet and clean it by hand. Easy peasy, and inexpensive too, the fitting costs about $5 at most well stocked homebrew supply stores.