Not sure. They did that for a lot of people when the Ring came out but I'm not sure about other cards. I love their customer service. You could call them and ask. They are usually super nice and very helpful.
You can call and ask, though your concern about that drawing attention to the MC is noted. However, if the consensus here ends up being "Yes, Barclays usually will do that", when you call to have it done you will be drawing attention to the MC. The only way you don't draw attention to the MC is if you don't call them.
Why are you concerned about drawing attention to the MC?
call the recon number on Monday.......They did it for me on my NFL card they tranferred the 1000 over to my Travelocity Amex when I had one....no issues
Took literally 2 minutes to transfer VISA CL to MC, so awesome!!
So did you just go ahead and close down the Visa?
EDIT: Had to re-read this thread....
And now you have a 7.2k card, too - love the higher CLs...