Now you see you are using a relay which was my first instinct but I had no idea how I could use the LS150's AC output to use as the signal to a 12 V relay or pair of relays , the manufacturer(Seep) provides very little info but there is this from a company that retails Seep products ,
https://www.gaugemasterretail.com/magento/seep-gmc-em1.html , it states 18 -25 V AC or DC , I have read from articles online that in practice they work best with DC at 18-20 volts and I have used a 20 volt DC supply to power one manually with great success , people have written that on AC they can hum annoyingly and the lifting action on the coupler is not as smooth as DC , your drawing shows 15VAC but if you only use the common terminal and + terminal on the LS150 you get a very low voltage whereas if you use 2 diodes you get around 8 volts half wave DC .