Lens errors, at least on DIGIC <= 5, come from the MPU. We don't know how to suppress them yet - I tried to work around ERR20 several years ago, but it didn't work.
I happen to have a faulty lens that also gives ERR01 and only works wide-open - so I took a quick look. Apparently, the MPU message related to this error is:
*** mpu_recv(06 05 03 29 01 00), from ff1bf420 ; PROP 80030023
PropMgr:ff01300c:8b:16: DispError 1 ERROR Time 2020/11/10 13:42:40
The 60D appears to ignore ERR01 and continue shooting after half-shutter press, but the 5D3 does not. So, at least on 60D, it might be possible to somehow block that MPU message in order to block the error. However, I had no success doing that - display goes black and camera becomes unresponsive, but background tasks are still working.
Disabling the error logs saved by ML is straightforward, but won't help much, as these logs are saved in background, without delaying normal operation. Feel free to edit the source code (debug.c, ErrForCamera_handler) and customize it as needed.
Edit: after pressing half-shutter on the ERR01 dialog, what happens is that camera actually reboots. As such, ignoring this error must be done on the MPU side - possibly by reprogramming it. Short answer becomes: not possible to ignore this error.
The only workaround I can think of, is to use the camera without ML, so the camera can reboot as fast as possible...