I am not commenting because I have any expertise, but because I am not sure the other suggestion is the best or only option....and I certainly agree that you do not want a roof penetration.

I have used this flooring in basements that get moisture and its great. 

I would think having a bathroom vent going into the roof would cause moisture, which could result in even more problems. 

Find the source of the moisture.

The biggest problems I've seen with hoarding houses have been related to moisture damage and mold.

The mold contractor recommends taking care of the moisture/dampness issue first, OR just get a dehumidifier that drains into the sump pump because it is very normal for new jersey basements to have moisture in the basement.For waterproofing: from speaking to different contractors, I was told my options are to install a french drain (best quote was $4,800) or another option was to escavate the exterior of the foundation to waterproof it from the outside, and re-grade the gravel on the exterior of the foundation (VERY expensive >$10k), or the third option is to just do the dehumidifier and drain it into the sump pump as the mold contractor suggested.The buyers still want the house, but I don't want to spend a ton just to finish this out. 

As far as moisture, I look on the wood for white residue as a sign for concern but I don't see anything like that in your pics.

Solar fan:PRO: doesn't run on electricity, no rodents, no moisture, fresh air CON: expensive $599 (covers up to 1200 sq ft) to $899 (covers up to 1600 sq ft) installedRooftop fan:PRO: cheaperCON: requires power and thermostat setting/can increase utility bills

I don't know the KC area, but here in Florida we've got so much moisture and termites these old houses can look fantastic sometimes, but behind the sheet rock are rotted away, and there will be plenty of deferred maintenance. 

Tomorrow will be a quick check on the moisture, punch a few holes in the drywall to circulate air flow and change the locks.