Good news! Amazon Store Card approved me for a 3.5k increase and Costco AmEx approved me for an 8.4k increase, so now my overall utilization dropped from 65% to 41%. Nice! As far as I can tell they were both soft pulls.
Thanks again to everyone for your information, feedback and advice, it's been very helpful. At the moment I don't qualify for much in the way of balance transfers, so I'll focus first on paying down my highest APR cards (the Capital One cards).
P.S. Apologies for using incorrect terminology in my OP.
