The smaller cories are very active generally. What are the Peppered cories you have? are they Habrosus or are they the bigger "Salt and Pepper" cories which scientific name I do not know. Habrosus (Dainty), Hastatus (Dwarf or Pygmy) and Pygmeaus are three of the smaller cories, Hastatus being the smallest and more of a mid-level swimmer than a bottom dweller. All three are very active and like to be in tanks that are 20 gallons or bigger (preferably longer than taller). All corydoras are wonderful companions to Betta's as they occupy generally different levels of the tank although the Betta's tend to go all over the place
Unless you have an extremely aggressive Betta(s) that don't get along with anyone, Betta's generally get along very well with corydoras of any type.
How many Peppered cories do you have? Perhaps you need to increase their numbers if they are hiding all the time; they may not feel very safe is all.