For the Garde-du-Corps, a total of 116 all ranks was the establishment. Actually the unit was staffed as follows, presumably for cost reduction, until at least 1815.
Großherzoglich badischen Garde du Corps Eskadron6 officers
9 non-commissioned officers
2 trumpeters (1 in reserve, without horse)
10 carabiniers
50 gardes (10 in reserve, without horses)
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77 total (of which 66 mounted for service)
So although called a "squadron", the unit was really the size of a company.
Commanders were : 1800 Obristlieutenant v. Münching, 1806 Stabs-Rittmeister v. Baumbach, 1814 Major u. Flügeladjutant v. Geusau
They did not leave their garrison in Karlsruhe, except in 1815, when they went to the seige of Strasbourg.
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The website linked above has confused the Baden Life-Grenadier-Guard Battalion with the Life-Grenadier Regiment. The latter, of 2 battalions, did have the extended combat record listed at the website. Their regimental history link
For the unit that was asked about
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Großherzoglich badischen Leib-Grenadier-Garde Bataillon
The Inhaber was Großherzog Carl Friederich.
The commanders were : 1800 Oberst v. Stetten, 1806 Oberst v. Porbeck, Obristlieutenant u. Flügeladjutant v. Stockhorn, 1814 Obristlieutenant v. Beust
To my knowledge, the unit did not leave its duties guarding the Grand Ducal palace until the months of January and February 1814, when they attended the battles of Brienne and La Rothière, standing in the reserve.