Yes, you realy stir this pot.
I do not believe in Bowden's figures. It was his estimations and these also could include mistakes.
Therefore I am always sceptical to every numbers which are put without truly historical reports.
Problem is that even historical reporst many times did not show real truth.
That is way I cast only these c.a. numbers which were showed by many historians as basing information.
You could check some more Digby Smith books. It looks that he could be closer to this real numbers.
If Allies had lesser men they would be waited for reinforcements. Russian forces were on the way and Archduke Charles also had big army in Hungary. Then they simply could and should waited. But they attacked it could mean that they belived that they had somehow advantage.
After so many battles lost, they knew very well French good quality valour.
Therefore Allies could not reliant on their quality, however they had almost two times more Guard soldiers.
Then their advantage must be in their numbers I believe. That's why they decided attack.
But at Austerlitz even 13,000 men more were not enough. Napoleon's superb strategy destroyed Allies which clearly went into his trap. They poorly manoeuvred while they sent few columns on French right flank, it led to some troubles. French attacked on left flank and in the center and Allies tried closed big gap there. This way they must used their reserves. But Russian Guard and Austrian cavalry can not stop French.
Then Allies started to retreat and it led to chaos and even panic. Some soldiers run away. Tzar did it himself only with few his companions.
Problem was that Allies left flank forces were in narrow place now. They must retreat crossing softly frozen lake and boggy terrain. French artillery easy can bombarded crowded Russian. Therefore there were so many casualties and losses.
These facts and not the only numbers, could explain overwhelming victory of the Emperor at Austerlitz.