This reddit post from over 10 months ago pretty much outlines the precise problem I'm facing: https://www.reddit.com/r/rainmachine/comments/xqgx5k/local_lan_direct_access_issues_with_android_and/

The local access is horribly slow or completely non-responsive and times out. After a reboot it works for a bit then it reverts back to being completely inaccessinble. It is obviously a legitimate bug that does not appear to be getting a resolution. Fortunately I very seldom have to interact with my RainMachine as it tends to just do its thing, but now that I integrated it with Home Assistant I was looking at it a good bit closer.

I have a Touch 16 with 4.0.1144 firmware.