I've chosen to use Light Dragoons as my Horse unit. The numbers represent the Infantry Battalion, Company of Horse, and Artillery Battalion. The three by three format works out perfect too as I generally use 3 Armies in my campaigns, meaning there's one whole Infantry Regiment (Royal Scots, Queen's Royal Reg., The Buffs) in each Army and a whole single Regiment each of Dragoons (Royal Dragoons) and Artillery (Royal Artillery Reg.) spread equally throughout the 3 Armies. Brilliant! These names also fit within the 32-character limit of units (unless the Regiment name is too long, then I just abbreviate it).
1st Army
A Company, 1 Royal Scots - Line Infantry
B Company, 1 Royal Scots - Line Infantry
C Company, 1 Royal Scots - Line Infantry
A Company, 2 Royal Scots - Line Infantry
B Company, 2 Royal Scots - Line Infantry
C Company, 2 Royal Scots - Line Infantry
A Company, 3 Royal Scots - Line Infantry
B Company, 3 Royal Scots - Line Infantry
C Company, 3 Royal Scots - Line Infantry
D Company, 1 Royal Scots - Grenadiers
D Company, 1 Royal Scots - Grenadiers
D Company, 2 Royal Scots - Grenadiers
D Company, 2 Royal Scots - Grenadiers
A Squadron, 1 Royal Dragoons
B Squadron, 1 Royal Dragoons
C Squadron, 1 Royal Dragoons
A Battery, 1 Royal Artillery Reg.
B Battery, 1 Royal Artillery Reg.
C Battery, 1 Royal Artillery Reg.
General's Unit
2nd Army
A Company, 1 Queen's Royal Reg. - Line Infantry
B Company, 1 Queen's Royal Reg. - Line Infantry
C Company, 1 Queen's Royal Reg. - Line Infantry
A Company, 2 Queen's Royal Reg. - Line Infantry
B Company, 2 Queen's Royal Reg. - Line Infantry
C Company, 2 Queen's Royal Reg. - Line Infantry
A Company, 3 Queen's Royal Reg. - Line Infantry
B Company, 3 Queen's Royal Reg. - Line Infantry
C Company, 3 Queen's Royal Reg. - Line Infantry
D Company, 1 Queen's Royal Reg. - Grenadiers
D Company, 1 Queen's Royal Reg. - Grenadiers
D Company, 2 Queen's Royal Reg. - Grenadiers
D Company, 2 Queen's Royal Reg. - Grenadiers
A Squadron, 2 Royal Dragoons
B Squadron, 2 Royal Dragoons
C Squadron, 2 Royal Dragoons
A Battery, 2 Royal Artillery Reg.
B Battery, 2 Royal Artillery Reg.
C Battery, 2 Royal Artillery Reg.
General's Unit
3rd Army
A Company, 1 The Buffs - Line Infantry
B Company, 1 The Buffs - Line Infantry
C Company, 1 The Buffs - Line Infantry
A Company, 2 The Buffs - Line Infantry
B Company, 2 The Buffs - Line Infantry
C Company, 2 The Buffs - Line Infantry
A Company, 3 The Buffs - Line Infantry
B Company, 3 The Buffs - Line Infantry
C Company, 3 The Buffs - Line Infantry
D Company, 1 The Buffs - Grenadiers
D Company, 1 The Buffs - Grenadiers
D Company, 2 The Buffs - Grenadiers
D Company, 2 The Buffs - Grenadiers
A Squadron, 3 Royal Dragoons
B Squadron, 3 Royal Dragoons
C Squadron, 3 Royal Dragoons
A Battery, 3 Royal Artillery Reg.
B Battery, 3 Royal Artillery Reg.
C Battery, 3 Royal Artillery Reg.
General's Unit