Bleach is readily available, very inexpensive and works well. Again the amount to use is 1/4 cup household bleach for every 15 gallons of water. Most of us overdue the soak time. The minimum soak should be at least an hour.
More is not necessarily better and eventual high bleach concentrations for long periods of time might kill pump seals. The RV should handle the recommended protocol with no issues. My pump lasted 10 years including 2 years with full time use. I replaced it for about 15 minutes of effort and less than $100.
Be sure to treat the hot water system including the HW tank. That means you must drain it, fill with treated water and then drain again after an hour or so.