As you haven't selected a flight director vertical mode, the flight director is still in take-off mode, engine failure operation will not be affected.
Engine failure for the flight director commands V2+20 in anycase, after achieving that speed.
Incidently, should you just reach over and engage an autopilot (above 400'), then the bug does the same thing - it jumps to the V2+20 knot position to match the Flight Director, but in addition the pitch mode changes to LVL CHG and pitch mode FMA is MCP SPEED. "Bug one click" keeps the FD in take-off mode, with take-off power.