Here’s how I get the bleach into the tank; we have the same setup as you. The inlet is the same as ours and it is the only way to get water into the tank. There is a gate valve that sends the water from the hookup either to the fresh water system or to the tank.
We built a filter system that lives in the locker with the water hookup. To get the bleach into the tank we remove the second stage filter cartridge and put the bleach in the housing and then fill the tank.
Works great. Hope this helps.