Welcome fella's.
Unless you plan on using your Omega for cooking breakfast, there is not much point in "seasoning" it. Afterall, it's not a cast iron frying pan.
Not real sure about #13 cleaner other than T/C seems to sell alot of it.
Dish/Laundry soap and hot water pump-flushing until the barrel is hotfollowed by alcohol patch and a dry patch. Set aside for a few to allow for the moisture to evaportate.
Lightly lube with a quality gun oil (Otis ultrabore, Clenzoil, B/C Sheath, etc)inside and out.
Remove the oil in the bore prior to shooting.
Don't need to remove the trigger assembly. Spray with a moisture displacing lube such as Birchwood Case Sheath. Wipe away excess.
M2C,
Tahquamenon