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What people usually do is call in and see if you have any offers to lower your APR. Hopefully, they'll get you to someone past a front-line CSR, as they seem to have a lot more options available. Last time I called in, I went from CSR to Analyst and finally to Retentions. Odd enough, just as I was about to close my Rewards and consolidate it into my CashForward, they magically found an SP CLI and a 0% APR offer.
From MY experience, i seem to be getting SP CLIs twice per year.
It seems like Barclays is a lot like Chase when it comes to giving out CLIs: They will give them without a hard pull, but they'll never do SP CLIs on request (if you request a CLI, it WILL be a hard pull).
I know from experience that Chase tends to give CLIs if you tend to use up a large portion of your credit line and you PIF, and they tend to do so every 6 months or so at most (I'm not sure PIF is a requirement, but that's all I've ever done
). The Target REDcard is also the same deal, except they won't let you request a CLI at all (not even if you want to take a HP). I doubt Barclays is much different.