its not always about the speed.
staysail is a sail which aids in balance through the range of conditions, aids in handling the boat in tight quarters, and also with the right headsail can improve speed. if you are using a 135 genoa then the staysail should likely stay down. if your 135 is yankee cut as in that photo, then the staysail will add a lot of power. the subject is very complicated and I could fill a book explaining it - youd be better to buy one of the many written on aero-hydro-dynamics of yachts.
keep the staysail up (sail hoisted) and the headsail should tack through the slot fine. you dont have to try and tack them at the same time, leave the staysail till after your done with the headsail, it tacks really easy then....
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Brian Duff
BVI Yacht Sales, Tortola
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