Diagnosis
You're right to be careful. The outer dome has a weather-resistant coating, and using the wrong cleaners or cloths can smear that coating or attract more dust and water spots.
How to Fix It
Here's the shop-teacher approved method. 1) First, power down the camera by unplugging it. You don't want it trying to adjust focus or activate night vision while you're touching it. 2) Use a soft, clean microfiber cloth—the kind for eyeglasses or camera lenses is perfect. 3) Breathe gently on the lens dome to create a light fog. 4) Immediately wipe in a gentle circular motion with the microfiber cloth. The moisture from your breath is usually enough to lift light dust and grime. For stubborn spots like bird droppings or sap, dampen a corner of the microfiber cloth with distilled water (not tap water, to avoid mineral spots), wipe the spot, then immediately dry and polish with the dry part of the cloth.
If That Didn't Work
If there's a greasy film (from grill smoke or car exhaust), you can use a single drop of isopropyl alcohol (70% or higher) on the microfiber cloth. Wipe gently and quickly, then immediately buff dry with a clean, dry part of the cloth. Avoid any household glass cleaners, vinegar solutions, or abrasive paper towels.
Heads Up
Never spray any liquid directly onto the camera. Always apply it to the cloth first. And make sure the camera is completely dry before you plug it back in. Check around the edges of the lens dome and the housing seam for any moisture before restoring power.