I don't have a push button start but I do have a 2012 with the DCT. I had similar behavior with my vehicle when starting with the key. It had a noticeable pause and for a while I thought it was in my head as it was very brief at times but then progressed up to a couple of seconds. Not long after that, I was coming to a stop at a stop sign and the vehicle totally died. I coasted to the shoulder and tried to get it started again with no success. I ended up having to have it towed to the dealer and they found it to be a faulty transmission control module (TCM). Fortunately, it was still under warranty so it was totally covered.
Not sure if this is what is causing your problems but it sounds suspiciously similar and it could progress into something that can leave you stranded so you might want to have it checked out.