I'm also experiencing this issue, where under the same conditions, if I start up the Twiddler 4 and leave it on the default config, it remains connected for at least five minutes, with or without typing - but if I start up the Twiddler 4 and switch to config 1, it disconnects after a few minutes, and does not reconnect until it is powered off and back on. I tested it on a Pixel 9 Pro and two computers. I haven't tried updating any firmware yet, but the Pixel is fully up-to-date, so the problem on that at least should not be firmware-related. The config I am using is Backspice converted using https://github.com/DDRBoxman/twiddler-cfg/, then uploaded to the Twiddler Tuner, tweaked, and then re-downloaded.
This is, honestly, a rather annoying first impression of this platform. But, it sounds like there is enough information here, and I hope the devs can reproduce and fix this issue. In the meantime, I will just use it wired.
Edit: I enabled Android Bluetooth stack logging to try to capture the BLE packets to see if I can figure out what's wrong, and with stack logging turned on, the issue is much less reproducible. I still get disconnects, but after an hour or so of use. On the other hand, when stack logging is turned on but in filtered mode, and in the filter settings "Leave only ACL headers" is checked, the problem goes back to disconnecting within minutes. This makes me think the issue may be timing-related, such that if the host is slower, the problem has a lower likelihood of occurring.