Sewer smell in bathroom ONLY when central a/c is on and kitchen sink is running water
This sounds like a similar problem I saw with a utility sink in a home.
You could see the gas bubble in the sink, so I made a larger trap about an inch deeper. It slowed the bubbling, but now another sink in a bathroom on the other side of the wall started doing the same.
When this was happening the house was developing slightly negative air pressure. I found the make up air duct (which is supposed to bring in fresh air from outside) had broken and was just pulling from the inside. A few air leaks in the attic and the home went negative and started sucking air through the P-traps.
I pulled the insulation that had been sucked into the duct, reconnected the flex pipe, and the problem was solved.
It sounds like you have a similar issue but it takes the additional water being added to the drain to cause the smell. I would add a make up air duct. They do waste a small amount of energy but make the home air healthier (even without the sewer issue). You may need the plumbing checked but if the house is positively pressurized this probably will quit happening.
That house had multiple sinks on the same vent that trapped air in the 2 back to back, this may be the same issue that a new vent or make up air would solve.