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In my experience with the CSP, it will grow--- slowly, but surely! I received it in April 2015 with a $5000 limit, and have received 2 CLIs to $13,000 now. It seems to follow other CLIs (sort of). My first CSP CLI came after a big Discover CLI, and the most recent one happened after my Amex CLI to where it is now.
I would never use a HP for a CLI, and YMMV of course, but I think the card should grow as your other CLs do.
Congrats! You shouldn't be that suprised since you only have one 5 figure CL at this time combined with your scores, $5000 would seem right. Use card hard, pif and pull the HP lever in 6 months and I bet they bump you to $10K or more!11