Thanks for offering different troubleshooting ideas, that helps me look for more work-around ideas and I think that I have found the "problem", although not the "solution" yet.
As I have used successfully the CL53L5CX on the mini JST connector, I just moved it to the regular I2C pins (D11 and D12) and both CL535CX and Serial2 now WORK TOGETHER, as expected!
I bought both CL53L5CX and CL53L1X sensors from Pololu, but it looks like that the Pololu's sample software for the CL53L1X is using I2C somewhat differently than the sample software for the CL53L5CX. So at least, I now know that I need to direct my questions to Pololu folks instead!
Thank you very much for your help!