Stirring may not have woken it up but racking to secondary may have re-started fermentation. The problem though is your yeast may not be able to handle anything more and therefore, re-pitching was a good idea. Do not worry that you dry-pitched. There are lots of threads discussing dry pitching and rehydration. There are always going to be various answers, but the last thing you need to worry about right now is that you did dry pitch. It may just take slightly longer to start given the gravity of the beer.
I would lower the temp in your house a bit. Remember, during active fermentation the temperature will rise slightly. While it is 70F in your house, it could be much warmer in the center of that bucket. You will develop off flavors if you're re-starting fermentation. Hate to see you lose a beer at this point, a good beer at that, due to off flavors because of heat.
So I have a fair amount of experience with a stuck stout and had spent a fair amount of time researching, discussing with experienced brewers, and just basically trying to get that stout to wake up. One thing is that you either should have started with 2 packages of S-04 or done a starter. Either or really at that point. Aeration is key in all beers but especially in the bigger beers. Did you aerate and if so, how and how long?
Can you link to the recipe you used? One thing about these darker beers is higher pH levels. Think about heartburn. If there are a lot of darker grains that do not produce a lot of fermentable sugars then there is simply nothing else for the yeast to eat. My issue was just that. I brewed using extract but also had a large grain bill for my steeping grains. In fact, as I look at it now I think to myself how incredibly dumb the recipe is but oddly, it turned out amazing. It is borderline too sweet but not as much as anyone expected. I have shared enough to now know it is not a total trash beer.
So with that all being said (and I apologize for the length) you have little to worry about with the dry pitching. I would have indeed taken the time to check my gravity anytime I rack to secondary. I don't know if the recipe was really necessary to do 2-stage fermentation, but I avoid secondary as much as possible. I'm new and a worry wort (no pun intended).
Give it a couple of days, see if you can tell if there is any airlock activity. After it's been in secondary for 7 days, take another reading. If it is not moving at all, come back and perhaps we can see what we can do. Also, taste it. How does it taste? I mean, 1098 to 1040 is still over 7% ABV but of course, we want to avoid bottle bombs.
FWIW, I did bottle my stout that started at 1067 at 1030. So far so good and they've been in bottles 4 weeks and have been shipped to a few places for people to try. Maybe that will give you some hope.
Side note, for my own learning opportunity, did you use a blow off tube? You noted the lid coming off twice so I wonder about that often.
Again, sorry for the length.