I want to just quickly add a data point here in case others find it useful:
Yesterday Swiss changed my flight on 24/8/2020 from a codeshare (operated by SQ) SIN-ZRH and then LX to VCE, leaving at 01.25 from SIN. It was changed to LX metal flight leaving 24/8 evening at 23.30. I did not accept, then was asked to call Swiss. I did so, with alternative flight already at hand (00.30 - SQ SIN-MUC, LH MUC-VCE). Process was quick, friendly service and no problems. When ticket re-issued, business had been bumped up from P class to D class (LX ticket stock).
I found service to be very friendly, competent and willing to accommodate specific requests despite fare and fare class differences. Granted, I was switching to other LH group codeshare, but I had the feeling if I had asked for any combination of flights within LH group it would have been no problem.