Do I need to render a salute to an officer when my hands are occupied?

Yeah, the coffee cup is not the best reason to not salute… if you have to be in those situations and absolutely must have your coffee cup, then get a shoulder bag or something that you can put binders a paperwork in, carry your coffee in your left hand, and always keep your right hand free.

That’s basic training 101.

Make a shelf with the binder in the left hand and place the cup on top with great balance, then render the salute. Some people carry a binder and/or other objects in both hands solely to avoid having to salute. In very few cases, some just feel self-entitled and want to pump their chest.

FWIW (one in a long family list of military service): if easy to do move everything to your left to salute. However, if you’re obviously carrying/moving things, and getting work done, and anyone complains you didn’t salute they are wrong. Look me up in the GAL I’ll happily see them to see if they qualify for intellectual disability and need to be separated…

Sounds to me like you are contriving a way to avoid the salute. Simple solution. Don't load up both hands unless you are carrying duty work. A coffee mug isn't that. If I were placed in that position I'd use the binder like a tray, place the mug on the tray, and salute. There are traditions and courtesies of the military services. Running to the AR to get the answer misses the whole point.

Learn to set your mug on your binder.
Sorry, but for some, it would be better to drop your coffee, than not saluting.

Granted, it is another story when in the field or carrying equipment.
If you can make the effort.
In the field, you are "not supposed to", but sometimes you run into that individual officer.
In the field, the greeting of the day is always "Sniper Check".
You may get chewed a little, but the officer usually gets the point. Usually.
Remind him, that for his safety, it is not a good idea to salute in that environment.

Of course everything changes, when the guys with the stars start walking around. Then common sense does not apply.

I do it all the time. Put the binder under your left arm and hold your coffee in your left hand. Render a salute.
It sounds, and probably appears, that this is the old "I had a thing in both hands" excuse. I suspect you are smart enough to figure this out.

If in their assessment(and you SHOULD know as well), that those items can be carried with reasonable comfort in your left hand, or shifting from your right to your left can be effectively done, you better do it.

There is no right answer to a power play. My suggestion is if you know this person to me rank narcissistic is to always salute with everything down on the ground and a formal salute rendered. That said you can always turn the binder vertical set the mug on top tender you salute retrieve your mug and carry on. Me, i would cross the street make no eye contact so as not to dignify “it” with a salute.
IMHO he knows..! Those who seek a salute know the never earned the respect of it so must demand it. If he earned your respect I’m quite sure you’d cross the street to salute a respected person of rank.
What ever you do don’t piss it off they’re carrier ending bite can be hazardous to promotions.

Your CSM is an utter asshole. If reg's specifically state you're not required to salute, when carrying something in both hands, then refer him to the appropriate regulation. This is the difference between the Army and USMC. In the USMC you not required to salute indoors unless under arms (i.e. on "duty").

Best unoccupy your hand, might not go well