Check the manual, but the default #1 boot device in BIOS may be the DVD, so if the battery is weak and not holding your settings, it would revert to booting from the #1 default device (eg, DVD).

CR2032 3-volt lithium ion coin cell battery, ~$2 at discount stores.

Thanks!  I think the big question is whether a new CMOS battery will fix the problem, or it is something else; but putting in a new 2032 battery is probably the easiest fix to try.
What is puzzling is why it fails on Restart but not after power-off, which is the opposite of what I would expect; but that may be unanswerable.

> CMOS battery removed for more than 30 seconds will reset the BIOS

That reminds me of another question; the manual says essentially "write down every single CMOS setting before you change the battery" which would be a massive task. Is there a core set of values I should jot down to save time and effort?

Ron's suggestion looks good. But, before changing the boot device replace the CMOS battery, as CMOS battery removed for more than 30 seconds will reset the BIOS, meaning again the boot order needs to be set.

Not necessarily you need to write down every individual setting. You can write down only the individual settings modified by you. Rest will take the default values automatically.

For example, change in boot device order, needs to be changed again after replacement of the CMOS battery.

You can always take digital photos of the BIOS screens. Just turn off the flash so it doesn't "white out"the images from the reflections.

Good idea; thanks!  (Have already photographed the backplane so I know what cables go where,and why.)