Half Bridge Controller not working
With the IRS2186 PWM control inputs have to be provided for both the high side and the low side.  This means you have to worry about things like proper phase control and dead time relations for the top and bottom FETs.  If not done properly, the FETs could cross conduct causing their destruction.  It is up to you to provide complimentary PWM high side and low side inputs with adequate dead time to the IRS2186 for half bridge control.  Usually using a part like a IRS2104 would be a better choice for half bridge, because it would take care of all the dead time and alignment issues for you.  IRS2186 is fine, but there is just more that you have to control.

In a half bridge the source of the top FET is connected to the drain of the bottom FET.  During operation the IRS2186 bootstrap cap is charged through the bootstrap diode with a return path through the bottom FET.  So, to charge the bootstrap cap, the bottom FET has to be on.  If the return path through the bottom FET is not made (bottom FET not turned on or something) there will be no bias for the floating driver because the bootstrap cap will never charge.  Make sure that during each 32kHz switching cycle the lower FET turns on.  Also, since this is a half bridge make sure that the maximum PWM duty cycle for each FET is less than 50%.  

A 0.5uF cap for the bootstrap cap should be enough.  You will also want a decoupling cap of at least 1uF at Vcc of IRS2186.  

Schottky diodes are not usually used for bootstrap diodes, since they don't have high enough reverse voltage for the application.  It is common to operate a half bridge off line with an input voltage of close to 300V.  \$V_{\text{RRM}}\$ of the diode will need to be \$1.5 \left(V_{\text{cc}}+V_{\text{in}}\right)\$ .  But, if your input voltage is low enough a schottky could be used.  For example for \$V_{\text{cc}}\$ and \$V_{\text{in}}\$ of 15V, \$V_{\text{RRM}}\$ of 1.5(15+15) = 45V would be fine. So, in that case a MBR160GS would work.  

\$V_{\text{cc}}\$ of 12V is a bit low, after subtracting about 1V for \$V_{\text{B}}\$ ripple and forward diode drop of 0.7V you're down to 10.3V which is really close to the UV lockout. It would be better to have a volt higher \$V_{\text{cc}}\$, or a larger bootstrap cap, like 1uF (less ripple), to have more margin for UV lockout.  

You could check operation of the floating drive by temporarily connecting \$V_{\text{S}}\$ of the IRS2186 to return and driving HIN with 50% PWM.  You wouldn't want to drive the transformer during the test, but you could see if the bootstrap voltage was good and that the FET drive worked.  Of course, you will need a scope.