First off, there is no way he could let you in the cockpit, if that's what you were asking. Generally the captain has to follow company/FAA procedures unless there is an emergency.
The CA does have some latitude when it comes to DENYING jumpseaters, (ie he may not need a good reason to deny a jumpseater).
However, if you do not have a JS agreement with SKW, then the CA would be violating company policy by basically giving away a free seat to a member of the general public...in the 121 world this is considered "nonrev fraud" and will get you fired immediately at most companies. Basically, it is like stealing as far as the company is concerned...kind of like asking the CA to go open up the company safe and give you $300.
In some cases you might get lucky, I have seen 135 and even fractional types talk their way into a cabin seat...but in this case you MUST have the gate agent's concurrence. She has final say as to who gets on the airplane...if you're going to bend the rules you would need her to go along.
Nect time try bringing cookies for the gate agent.