Insecticide will kill things like fleas,moth,beetle but won't remove allergens.
I suffer with allergies, and on most jobs with pets present I immediately get symptoms like sneezing, wheezing, eyes streaming and itchy throat.
Usually as I proceed with each stage of the clean my symptoms relieve. Dry soil extracting with the sebo, followed by a suitable pre spray (usually spm, citrus or whatever I'm using) agitate and dwell, followed by a Thorough rinse.
In most cases it's something in the dust which causes allergies, I'm no expert but the cat itself is fine, (hair/skin cells/oily dust) thats what come off the cat which is the allergen usually.