Oh how lovely! I can't stop smiling!
Of course Jeeves would come up with an idea to get them out of trouble - and I rather like the idea. Now they MUST get a cat, so that Bertie's friends don't get suspicious next time they visit ;)

Also, this:
"But he assured me that, to him, it seemed more appropriate to call me by these names instead of Bertram or Bertie."
I totally agree with this and it's my personal headcanon as well!