Laundry tubs backing up

Hello. We have been having a problem lately with one of our sewer vent stacks. It is at the back of our house and attached to our kitchen sink then continues straight down to the basement. The trouble is that it leaks whenever it rains. It doesn't leak a lot but it's a huge problem because it is right next to our electrical panel. We thought at first that it must be a roofing issue. We had a roofer come over and look at it and nothing appeared to be wrong but he suggested we seal around where the metal meets the roof. We did that and there was no change. Finally we did a test where we sprayed water on the roof and checked for the leak. No water downstairs. Then we sprayed some water directly into the pipe and that's when we saw the water downstairs within a few seconds. So we know for a fact that it's the pipe. The problem is we can't figure out where the leak is or how to fix it. It has to be somewhere above the sink upstairs right? Otherwise it would leak when the sink drains which it does not. It is almost impossible to get over to it from the attic because it is quite a distance and the attic is not high enough. We would have to crawl over there. If it was leaking in the attic would we see damage on the ceiling somewhere? My husband is thinking about buying a chimney rain cap for it. Would that work? Is it safe to do that? We would take it off every winter since we don't have any issues in the winter. Thanks for any help!

Ive narrowed the source of the leak to occurring when water goes down the tub drain. If I leave water in the tub with the plug it is fine. When I remove the plug it seems like a lot of water comes through the ceiling. I have a cast iron claw foot tub and the pipes upstairs are still lead (I know, I know) We've had no issues with water leaking after flushing the toilet. However when the water comes through the ceiling after a bath the leaking comes through under the toilet area. Which has me wondering if it is just due to an uneven floor/ceiling and the water going to the lowest point or is the leak further along the pipe? I'm ready to open the ceiling but not sure where to cut and check for leak first due to knowing it happens when the water goes down the tub drain but seeing that the water actually comes through the ceiling by the toilet. So, where do I cut to look for the leak? Or am I missing a step or 2 that I should consider first. I'm an avid DIYér but plumbing is not my strong point and I need some advice.