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Thanks for the help, everyone. I did try calling the Cap1 "account specialist" line. It's just a call center for new applications. CSR said only the automated system can handle CLIs. So, onto the love buttons.
Short story is that between all 6 Cap1 and Synchrony accounts, I ended up with a net $4300 bump (79% increase) but it was mostly on the accounts I didn't care about. Ugh. And Cap1 was stingy as hell - only 10% on the open loop TLs. I guess I need to make this a regular thing since I've never requested a CLI before - they would just be issued (or not) automatically.
Congrats on the increases! Even if they aren't on cards you use often they can certainly help to offset balances or get higher starting limits in the future. I usually request soft pull CLIs every 60-90 days, depending on what my credit looks like at the time.