Favored arms of CS mounted rifle... is a difficult question at best. A lot would depend upon time frame and theatre of operations as well as the political influence of the Col in charge. Each factor would show different weapon availability.
Favored would likely be the P56, P58 or P60 but reality is that the men would have been issued what was available. So Lorenz, any Enfield pattern arm, M1841 or literally any carbine they could get their paws on. But again the reality would likely be an issue of P53 arms with a lucky few maybe getting P56's. It's been interesting for me over the years to see the variety of arms available and used by CS mounted troops of all sorts. Mounted Infantry or Mounted Rifles are a fascinating study. Several years ago I spent so quality time studying the Mounted Rifle units raised in Texas to fight the Commanche. Those men gave stellar service despite being regarded as red headed step children, at best, by both the CS military & Texas.
The reality on pistols and carbines is that there were never enough to arm the cav; and Infantry didn't get preference over the cav when it came arms. The mantra put forward by some of CS cav w/ four pistols, a shotgun, carbine and a sword is so much hooey.
I'm really not sure if that makes any sense or not... coffee, need coffee.