First, can you add as much battery as you wanted? Meaning, is there a technical or score-wise drawback to the amount of battery? Or mass?
Does the servo load increase with the increased battery amount? (Probably not).
The above considerations would require a precise battery sizing. Otherwise, taking a rough estimation and then buying one size bigger and smaller batteries would be the fastest and most pragmatic road to success.
If you do not want to purchase unnecessary items, you can calculate the energy consumption using power utilization as below:
1) divide the total operational time into phases. Like, phase 1. startup and waiting
Phase 2. Competition run
Phase 3. Wait for Xxx
Phase 4. Run-2
Phase 5. Finished and waiting
2) In each of these phases determine the power drawn from the battery, and multiply that with the amount of time of that phase.
3) And when you find the exact number, add 20% for uncertainties. This would help select the required battery.
Notes: 1) temperature is ignored (assume the competition is indoors).
2) validating this method can be done by literally testing it 2-3 times for especially the operational cases (competition runs).