Finally sorted this out: the problem happens when I connect a second Twiddler to the host, and then continues even after powering one of them off. This occurs on both Linux and Android, but works fine on iOS. It seems like the Linux Bluetooth stack goes nuts when the second device is connected.
BT module: 0a5c:2148 Broadcom Corp. BCM92046DG-CL1ROM Bluetooth 2.1 Adapter
OS: Ubuntu 20.04.3 LTS
kernel: Linux gnocci 5.4.0-86-generic #97-Ubuntu SMP Fri Sep 17 19:19:40 UTC 2021 x86_64 x86_64 x86_64 GNU/Linux