I teach bounce recovery training in the 777 at my carrier.
some problems to it:
1. Re-establish a "landing attitude" (~4 to 5 degrees pitch) to prepare for a possible second contact with the runway.
Especially in hot conditions with an underpowered aircraft, the situation will look more as a impending stall than a bounce recovery. Therefore giving pitch values is delicate. With 5 degs a -300 will fall out of the air.
For such situations, i suggest (not teach) to apply the GPWS "pull up" warning procedure. Because it's something we train quite often and will save your day.
2. MANUALLY advance the thrust levers to GA thrust limit
Give it all it has, again as per "pull up", don't chase any value.
3. When airborne and climbing, select TOGA
Note that it is difficult to press the TOGA switch when desperately pressing the thrust levers full forward, due to the angle. We should train to call the PM to press the switch, call out the FMA and engage the AT again.
The rest is as per normal GA.
Of particular note is that a SINGLE PUSH of TOGA on any G/A commands a thrust that will give you "approximately 2000'/min VSI) for passenger comfort. With both engines running, you have to hit TOGA twice to get true G/A thrust.
Not entirely true. You will get full GA thrust if the aircraft does not get 2000'/min. As this is ample to feel safe, a second hit is not important and should not be chased for.
For such threatening situations, what i often suggest is:
Don't go for modes, go for moves.
Move the levers and yokes yourself, you can check the automation/FMA later