3 new cards and a few CLI’s on my existing cards (which I haven’t gotten for the over 2 years since they’ve been opened) is too much to ask for?
As mentioned, everyone is different. I'd suggest that your limit might be set by what you intend to do after...
I just picked up four accounts in December, and four more in January. One lender in particular (Barclays) has taken AA, but no other lenders have.
...I've got no immediate plans to apply for anything else, so I'm sitting in the Garden letting scores recover.
I haven’t decided which cards to go for yet. My main priority are higher starting limits. Which banks would you recommend for an app spree? Current cards are with Amex, chase, Discover, cap1, and synchrony.
What is your specific list of cards and current limits?
Setting up your file for CLI is certainly a place to start. If your main goal is basically increasing credit limits to improve utilization, some banks have SP CLI. Those are always preferred and may end up getting you what you want.
If you don’t have a card in mind for features you want, all the more reason to wait and do some research. The 5/24 rule for Chase is a block for some people. Once you have 5 new cards in a 24 month period, no new accounts from Chase. Only a few exceptions to this rule.
The once-per-lifetime AMEX bonus rule means you want to time any AMEX apps for max bonus.