Prince of Essling is proably the best man to ask but as nobody has responded here is as much as I can tell you.
Line Infantry Regiment ‘Graf Kaunitz-Rietberg’ Nr. 20 (German, 1792-1815)
Regimental Designations and Colonels-in-Chief:1785-1826: Infantry Regiment ‘Graf Kaunitz-Rietberg’ Nr. 20(Kaunitz-Rietberg, Feldmarschalleutnant [later Feldzugmeister] Wenzel, Graf)
Colonels:1781-1792: Riese, Oberst Karl, Freiherr von
1792-1793: Linde von Linden, Oberst Josef
1793-1794: Gortz von Zertin, Oberstleutnant Johann [acting]
1794-1795: Bender, Oberst Joachim, Freiherr von
1795-1795: Frankenbusch, Oberst Prokop, Freiherr von
1795-1800: Richter, Oberst Josef
1800-1805: Murray de Melgum, Oberst Albrecht, Graf
1805-1809: Bresslern von Sternau, Oberst Josef [1]
1809-1814: Luxem, Oberst von
1814-1827: Demuth, Oberst Johann von
Subordinate Officers (1799):Pateck [Patek], Oberleutnant Philip [2]
Neumann, Major Wenzel
Walthor von Waldenau, Major Franz [3]
Muller, Major Franz (1805) [4]
Regimental Grenadier Division:1779-1797: Formed battalion with divisions from Regiments Nrs. 7 and 56
1799-1801: Formed battalion with divisions from Regiments Nrs. 22 and 29
1809-1810: Formed battalion with divisions from Regiments Nrs. 12 and 26
1810-1818: Formed battalion with divisions from Regiments Nrs. 7 and 56
Recruitment Areas:1782-1807: Sambor (Galicia)
1807-1817: Troppau (Moravia/Silesia) and Sacz-Jaslo (Galicia)
Headquarters:1771-1817: Neutitschein
Notes:[1] Oberst Bresslern von Sternau was killed at Hausen [19.04.1809].
[2] Oberleutnant Pateck was mortally-wounded at the First Battle of Zurich (04.06.1799). He died 13.07.1799.
[3] Wounded at Fussen, Oberstleutnant Walthor von Waldenau died in 1800 in enemy captivity.
[4] Major Muller died in action at Gunzburg in 1805.
Battle of Austerlitz
The Kaunitz Regiment Nr20 was part of the Austrian 4th Column under Kolowrat (see below)
Fourth Column
Strength: 23,900 infantry, 52 light and 24 heavy guns. Lieutenant-General Mikhail Miloradovich and Feldmarschall-Leutnant Johann Kollowrat
- Advanced Guard - Lieutenant-Colonel Monakhtin
- Novgorod Infantry Regiment (1 battalion)
- Apsheron Infantry Regiment (1 battalion)
- Archduke John Dragoon Regiment Nr. 1 (2 sqns)
- 1st Infantry Brigade - Major-General Wodniansky
- Novgorod Infantry Regiment (2 battalions)
- Apsheron Infantry Regiment (2 battalions)
- Little Russia Grenadier Regiment
- Smolensk Infantry Regiment (2 battalions)
- 2nd Infantry Brigade - General-Major Heinrich von Rottermund
- Kaunitz Infantry Regiment Nr. 20 (900 in 1 battalion)
- Salzburg Infantry Regiment Nr. 23 (3,000 in 6 battalions)
- Auersperg Infantry Regiment Nr. 24 (600 in 1 battalion)
- 3rd Infantry Brigade - General-Major Franz von Jurczek (or Jurczik)
- Kaiser Infantry Regiment Nr. 1 (1,000 in 1 battalion)
- Czartoryski Infantry Regiment Nr. 9 (600 in 1 battalion)
- Lindenau Infantry Regiment Nr. 29 (400 in 1 battalion)
- Württemberg Infantry Regiment Nr. 38 (500 in 1 battalion)
- Kerpen Infantry Regiment Nr. 49 (700 in 1 battalion)
- Reuss-Greitz Infantry Regiment Nr. 55 (600 in 1 battalion)
- Beaulieu Infantry Regiment Nr. 58 (500 in 1 battalion)
- Vienna Jäger (300 in 2 companies)
- Pioneers (340 in 2 companies)
- Artillery
- 4 Artillery Batteries (each 6 guns)
- 2 (Russian) Artillery Batteries (each 10 guns)
- (Russian) Heavy Battery Nr. 3 (12x 12-pdr)
- Heavy Battery Nr. 1 (6x 12-pdr)
- Heavy Battery Nr. 2 (6x 12-pdr)
The fourth column was position behind that allied right flank during the battle. (see Map)

It took part in the defence of the Pratzen Heights and in the final rear guard west of Austerlitz during the retreat from the field.
Uniform
Regulation German infantry uniform with Lobster Red Facings and white metal buttons