Hi,
The short answer is no, there isn't a volume-preserving convert to mono in WavePad because it does as you guessed: it averages the values from both tracks. Paste Mix takes a different approach, adding the two signals and then peak limiting when the combined signal exceeds 0dB. But internally WavePad uses 32-bit floating point samples, and when we add these and divide them by 2 there should be no loss of precision, so no distortion. On the other hand, the paste mix approach may cause distortion if both signals exceed -6dB at the same time. The peak limiter does mask this distortion, but you still get a loss of information.
My suggestion is to use the existing stereo to mono conversion, and then amplify or normalize the file to your desired level. I know you said that this introduces distortion and exaggerated noise, but I can't think why this would be. If this two-step process is too tedious to perform on many files then you could use the Batch Converter tool. I was going to suggest that you could use the Batch Converter for your split-and-paste-mix process, but unfortunately that won't work because the Batch Converter doesn't allow splitting files into their component channels. You could at least set up some keyboard macros to make this easier.
Alex