sunken living room - help!

In typed:

Plus, if the original contractor comes back out to try to figure out a way to do the fix, they will definitely be charging you for that anyway. That part of the job is not the same thing as putting down the flooring, and when they were there they said they couldn't figure out a way to do it. Putting down the flooring is a different type of task than figuring out the baseboard situation. They know how to put the flooring down, but they don't know how to solve the baseboard issue.
Anyone with good carpentry skills could probably easily figure out a solution and do it for you. I have a multi-skilled contractor/"handyman" that I use for many projects. It is really just a job where someone needs to create the right baseboard trim around the room as others have described. And, just as an example, the person that I know could easily do this and his cost to me is only $35/hour plus materials. I use the term "handyman" for him because he can do a lot of different things well. If you know of such a "handyman"/woman in your area, he/she may be able to fix this for you.
You wrote earlier, "...I like the idea of adding a larger baseboard and chipping away to make room for the crazy concrete."
I think that is basically all that you will need. Whether chipping off some parts of the concrete that stick out, or back-cutting the new baseboard in those areas will be easier, will just depend on what you have there. The person who does the job may do a little of each.
Good luck.
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