RGB values change saving png to jpeg

When you look at what has happened to those images it is perhaps surprising that they come back with only a difference of 1 in one RGB channel, bearing in mind that the Save As and Save for Web encoding processes are likely to have different internal coding of the functional steps below and therefore different errors introduced to the encode process.

Jpeg is not designed to return a round trip of exact numbers (other formats with lossless compression are) - but is designed to give a perceived similar image following a quite severe reduction in stored image data.

Starting with RGB there is :

1. A conversion from RGB to YCrCb This is to a standard formula but will be subject to any rounding used in the calculations

2. Subsampling of the colour channels which effectively quarters the information in each of the two colour channels

3. A discrete cosine transform of each 8x8 block of data in each of the YCrCb channels, designed to separate the information crucial to definition from that which is almost imperceptible to the eye.

4. Quantization which is where the "quality" control comes into play and is where the lossy part of jpeg happens

5. Lossless encoding of the data

Decoding, when the file is opened, reverses all those steps, but as stated above any rounding in conversion will potentially introduce an error as will the lossy quantisation.

Dave

" select colour palettes which work on the print and digital colour values with no issues."

There is no colour palette or trick that you can use to stop JPEG changing colours. That's what it does, by design and absolutely deliberately, except for very rare accidents. That's why it's called "lossy". Your client needs to know this, just as you do.  Consider: why do you think there is a quality setting for JPEG? A quality setting would have no meaning if what it saved was exact. (No, maximum quality doesn't mean "no change").