How can I deal with duplicate file names when moving files?

Save yourself a lot of pain. Assuming your cameras are capable, customize the file naming in them, instead of accepting something like img_nnn, change it to CA_nnn, and CB_nnn, or something.

Also, rename the files upon import. My personal favorite is:

YYYYMMDD_hhmmss

(Primarily Landscape photographer, and that is a common syntax for the field, also tjat file name ayntax allows for easy file sort)

Oh, and Keywords, Keywords, Keywords.

/Edit/Followup/

I see your cameras might not have the custom file name capability. Ok, rename the photos at import. Pick a syntax like YYYYMMDD_hhmmss, or whatever you like.

I'll be honest, all this micromanaging of folders and moving-around of files sounds like a tedious nightmare to me, exactly the sort of task that Lightroom was invented to let us escape!

Some suggestions: an approach I personally find both robust and efficient: 

• develop an import scheme to have LR copy-import new images into computer storage automatically
• save this scheme as a named import preset, for exact repeatability
• Base this auto-filing hierarchy and folder naming on objective and unchanging attributes of the images themselves: something LR can self implement easily: their date of capture.
• It is then vanishingly unlikely that photos from two camera bodies that may be named the exact same, will also happen to get filed according to the exact same common date of capture. 
• Once imported direct to their permanent home, practicality strongly prefers: leave those files and folders be. The capture date of an image is one property that can have no need to ever get reconsidered: unlike e.g. subject based folder schemes.

IMO manually micro-managing photo files and folders simply does NOT have to be any part of your day - or of your month, or of your year even. Put LrC in charge of all that, that it is what you pay it for. Get to where you can just relax about the whole issue!

Then you can devote (from the start) all organisational effort into virtual management by Collection, and also into keywording the images (an investment which then also enriches the output produced.)

The whole idea of moving photos from one folder to another folder is one that many people avoid. You can use keywords or collections (I prefer keywords) to identify photos and to allow you to search, and then it completely doesn't matter what folder the photos are in, all the ones you want to be grouped together are now grouped together by a keyword(s) or by a collection(s).

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@Bazsl wrote:

I realize that many hgh end cameras let the user change the file prefix and this would prevent files from two cameras having the same name. I also realize that I could dump the files from CameraB into a folder and use the Bulk Rename Utility to add a prefix to the files before importing them into LR. My question is, is there a way to handle the duplicates within LR. Thanks.

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If renaming is an acceptable option, you don’t have to move them first. You can have Lightroom Classic apply Library > Rename Photos to selected images in their current folder, having it include a metadata token that makes the filename unique. Some photographers have it add each camera’s serial number to the filenames, others add the capture date/time. Capture time might be safer because it’s possible for the same camera to record the name IMG_1234 more than once within a few years.

After they get their unique filenames, you can move them without a conflict.

Making the filename unique is part of my default import preset, to prevent conflicts later.