For what it's worth, when performing an LPV approach with my AFS/Dynon/GNC355 setup, I set minimums prior to beginning the approach. Once the glide slope is captured, I set missed approach altitude and continue to minimums. Unlike an ILS where the MAP (missed approach point) is determined by a baro altitude on the glide slope, LPV MAP is a waypoint (usually over the middle of the runway). My autopilot will level off at minimums and continue to MAP. Once reached, the Garmin navigator prompts you to execute a missed approach. If activated, my autopilot will begin a climb (at pre selected vertical speed) to the missed approach altitude. Additionally, the missed approach procedure and hold are activated.
I do not have a TOGA button.
Hope this helps.