Chase has the option to determine if you are preapproved for credit. It is a soft inquiry so it would be a good idea to check that to see if CSP comes up as an offer. Note that preapproval is not a guarantee of a credit offer. I will include the link; if the moderator removes this, their pre qualifier pops up on the main page with a picture of the CSP, Freedom and Slate grouped together:https://creditcards.chase.com/prequalified-offers?CELL=64QW&jp_aid_a=62153011&jp_aid_p=col_uk_home/trip3
They like to pull EX; I would wait the two weeks (keeping in mind that the creditors in question may wait until the account's cycle date or beginning of the month to update to the bureaus), pull your EX to see if everything is looking good and go for it. Many have posted that Chase does not like to be your first $5K + issuer (CSP is Visa Signature and requires a minimum $5K limit); others will disagree but that has been the general consensus. You may want to take advantage of the Amex 3x CLI benefit; try to increase your Amex to $7,500 and, if approved, let that report to the bureaus before applying for CSP. Just my advice.
Also, make sure you will have $4K spend in the next 90 days to qualify for the sign up bonus.