P: copying just the temperature and not the tint settings

If you just switch the conception away from the idea of syncing or copy-pasting changes you have already applied:

you can highlight the images concerned, turn on AutoSync, and then any new changes made thereafter are selectively and immediately applied to all the other images highlighted as well as to the active image. For example, say you select five images and the first of those is active. You can e.g. as appropriate choose "As Shot" WB and these will all be immediately set to As Shot temperature and tint, if they were not already, whatever that individually means for each photo. Then you can e.g apply a chosen new Temp value. This will set all the images to that new numerical value for Temp, leaving their various Tint values alone (which may or may not give the look you want). Note that this may be a big change in Temp number for some images, and a small change or no change in number for others, depending on their previous setting (which came from the selection of As Shot). This has been an absolute change, TO a certain new value. Also the way Temp works is different for a Raw and a non-Raw image so the same value is not applicable to both.

If what you want instead is to apply a relative change: to adjust Temp BY a certain amount, in Library is the Quick Develop panel. Here you can adjust images in a relative manner, offering the ability to nudge (some of) their adjustments up or down in small or large fixed steps.

In order to apply such a relative change onto multiple images in a batch, you will need to first highlight them and turn on the separate AutoSync that Library has its own toggle switch for - at lower right, left end next to Sync. Then experiment with the change, and results are seen on all these thumbnails as you do so.

Remember to turn off AutoSync again, IF you don't want that batch working mode to continue happening. Otherwise, if you leave it active, you will need to be constantly aware as you work, of whether you have currently got just one image highlighted, or many.

I suspect that this option is not available since white balance always is a mixture of both, the temperature and the tint.

I am not aware of any option to do that with a batch of images.