Reaching behind engines can be a pain. Often you can find tricks to get a washer and a nut onto a screw though. If not.....
The tough part here is to determine if and how much of the wood core has become wet and also dry it out. Usually I would recommend drilling test holes FROM THE INSIDE, into the core material but not all the way though the outside gel coat. You get core samples and can tell how far the water in the core has spread and can then easily just fill the test holes in less than 5 minutes when you get done. In your case, you may not be able to do this given the engine location.
If you really want to do it right, well, I think you see where this is going. If below the water line maybe you drill holes from the outside, since you can just paint over later. You could try a moisture meter, but watch out for the large number of operators who think they can just take an absolute reading on the guage. Or...There is an infrared type camera that does a GREAT job of seeing into the laminate. This might be the best option if you can find one. I believe it was Bob Austin who did this.
Or...you can just pull the engines, which if you can't get to the transom any other way, might be the answer anyway.
BTW, I thought I'd more publicly state that the opinions you have received to just "move the bracket" and leave the screws are downright scary. It's this level of incompetence that is the root of core failures. The fact that the manufacturer installed the brackets the way they did is near criminal. The suggestions of the inappropriate use of adhesives are equally scary. It's bad enough to pull stunts with 5200 on your own boat, but absolutely irresponsible to suggest such a thing on someone elses boat.
The good news? Since the boat is a 2007, you have a very very good chance that your core is nothing more than a little wet. Wet plywood can last a lot longer than you might think. When completely submerged it actually does not rot very quickly. It's when it starts to dry that problems start. When that oxygen gets in there and mold spores start to grow is when things go bad. When the moisture SLOWLY wicks to dryer areas and in return the dryer areas contribute air, that slow process is what will cause the mush effect. Which is why drying the core is so important. You really realy really don't want to just seal that moisture in. However a few core holes, a little time with a heat gun and a few days with a box fan will accomplish more than you might otherwise think possible. If the area is small, a few hours with a heat gun will actually do it. Anything that got wet or even a little damp I'd treat with CPES too, just to make sure. It will seal and kill any mold spores in the case that it ever see's a little water again, having those mold spores already present can accelerate a future failure. Once treated with CPES, it will be better than new if the wood structure is still good.
Now...the repair itself is not at all difficult or timely. The question for you is access to it. In years past I spent a lot of time working around engines, even dismantling key pieces such as manifolds to gain access to areas around the engine. That is, until I discovered that I could unhook an engine in less than two hours and the marina lift could then pick it right out of the hatch in less than an hours bill time. Depends upon each boat, but if your engines have easy hatch access, sometimes we spend days working around what otherwise is only a couple hours work in the mistaken belief that pulling an engine is always a big deal. Sometimes its the littlest deal.
