I'm not 100% sure but all pumps have to cycle on & off unless you have a gravity feed or air pressure style. Way back in my first RV days I had a small water tank above the sink and it was gravity flow. Also had older camper that air pressure 'pushed' the water out of the tank & thru the lines. We didn't have a shower back then though & only one sink. Now a days the water tank is in the frame below all the items requiring water. Water usually needs pressure to flow up hill. I use a Sur Flow Ultra Quiet. It is very quiet and works well. You may want to try one with an accumulator tank as well.

Adding an accumulator tank will smooth out the pump cycling.

My water pump cycles on an off when the water is on, that's they way it works. I understand that you can buy a higher quality water pump that's not as loud. I just had to replace mine but wanted to avoid any modifications to I purchased the newer model of what I had and sure enough I still get the off and on noise like the first one. If you do find a quieter one please post it if you purchase a quieter one.