Origin of RAF Motto

[Archimedes]

FP -
The AOP unit numberplates were RAF rather than RA, so they transferred from the RAF to the Army, rather than within it.

Short answer to your question - Probably not.

Long-winded answer to your question -

There could be a couple of issues about 678 taking on the badge and motto of 644.

The first is that if offical approval for the badge be sought, the College of Heralds would, I suspect, note that the badge (and motto) had already been awarded, and not approve the badge for 678 Sqn.

Second, I would imagine that 7644 (VR) Sqn's claim to the lineage of 644 would be a major issue. I'm not sure of the rules here, but if 2602 and 2603 Squadrons employed the old 602 and 603 Sqn badges, then it seems that 7644 have been a little remiss in maintaining the historical link; if 2602 and 2603 used different badges because - as the designation suggested - they weren't the 'real' 602 and 603, then the problem here is that if 7644 become 644 in another RAuxAF reorganisation, they'd then find that someone had pinched their badge....

The third issue is that there are still some 644 Squadron members around (there is a sqn website...) who would probably object to a new squadron being formed and simply taking the badge and motto without asking - particularly since the unit is (a) non flying [if I've read your description properly] and (b) an army unit.

Since 672 (for instance) wasn't awarded a badge when it was an RAF unit, it must have created its own. This would seem to militate further against a direct lift of 644's badge and motto.

As long as there aren't any expectations for 678 (if it is given that number) to adopt the badge of its predecessor unit (if there was one), then a new badge would need to be designed and it could pay homage to 644 Squadron's badge.