Yes it's that i do
but you have an internal "cap" on the first sphere and you must kill a part!

Else here my 3dm object in case of
https://manuelgrobotovf.weebly.com/uploads/1/3/5/4/13544449/2objects.3dm

Believed that all was automatic! :)

Edit : I am in the case of "Keeping" Trim

ok that works but you must CLICK one by one the Spheres and not "All " at the prompt :
"Select pieces to Keep or Push Done to Keep All"
Curious but working! Cool!

I have the 2 spheres with empty caps! :)

It's also working for my 2 objects but I must "Join" each object before run the Trim
And one more time that works not with the second "Done for All" but yes cliked "one by one"!
That was the trick! :)

So Perfect! : Join the Object A (solid) - Copy Move A = B (solid) (for future Trim with "keeping" )
Select the 2 objects A & B - Trim All (mutual) "Done" - Click Object A & Click Object B (and not Done for All keeping)
Then Done : et voilà :)