The transport department will send you a letter to your address if you don't have an address proof, which you then have to bring back to complete the application procedures. (That's how Mainland Chinese exchange their drivers licenses while staying in HK as a tourist). Maybe the library will do that too.
The most legal way, but not necessarily the simplest if for a helper to have a Hong Kong bank account. It seems that Hang Seng offers a special ATM savings account for only 500HKD to open, but then no minimum balance. Bank of China has a similar offering. And the latest news is HSBC has entered this game apparently starting August 1st, 2019 they'll eliminate minimum balance fees and make some over the counter services free as well - https://www.hsbc.com.hk/accounts/rem...-balance-fees/
Here is a link about BOC and Hang Seng, but you'd have to double check with actual bank representatives to confirm - What is the best bank account for domestic helpers?
Are you sure this is a real problem? You don't even need to be resident in Hong Kong to get a library card. Are there FDH's genuinely trying to use the public library system and failing due to proof of address?