Edited in recognition of closed-mindedness.

~~My brute force search shows no keys shorter than 10. Here are the only keys of length 10 and 11 respectively:~~

~~
~~RVLVQRVLVQ
VRLVQVRLVQ

VLRVQVLRVQQ
VLVRQVLVRQQ

~~Curiously, there are no keys of length 12. ~~

The only word of length 7 that does not crash under iteration is RRQRRQQ, and it evolves unboundedly. There are 74 words of length 8 that grow to over 30 letters, I think none of them cycles, and there are two eventually cycling words, one you gave and the other its pair RQVRLVQQ. The first time an odd period greater than 1 appears is at length 12, these are the originating words (all end up with period 3):

RQQVLLRLVQQQ

RLQVLLRLVQQQ

RQLVLLLRVQQQ

RQVLLLVLRQQQ

RQLVLLLVRQQQ

RQLVLLLRVLQQ

RQVLLLVLRLQQ

RQLVLLLVRLQQ

RQLLVLLRLVQQ

RQVLLLVQQRQQ

RQVLLLVLQRQQ

RQLVLLLVQRQQ