I'm not sure why we're assuming that flap deployment will cause pitch instability. As far as I'm aware, flap deployment moves the CoP rearward, which should make the aircraft more stable. I may be missing something here, though, I'm still working my way through Perkins&Hage. I'm not quite following Peter's argument about gusts.
For now we'll work under the assumption that extended flaps will cause a rearward shift of CoP in the pitch plane and thus increased pitch stability. However, this same rearward shift will intuitively, as you noted, cause a nose down moment, which would be aggravated if you did some kind of Fowler flap setup. This is corroborated by the above linked articles and pg.29 (pdf pg.33) of this NACA data.
Regarding your ideas to cancel this moment out to allow yourself to actually use the high cL you get from these flaps:
- Ducting the jet exhaust to the front of the craft: While within the realm of physical possibility, this would likely be a highly inefficient design due to the number of sharp turns the exhaust would have to take, and the amount of heat energy it would lose while doing so. Overall, it wouldn't likely be implementable.
- Additional engine: likely weight prohibitive.
- Fan driven by shaft: The most reasonable of your concepts in my mind. As you noted, it's been done before. However, as far as I'm aware, the Harrier uses ducted jet flow rather than a fan. I think you're thinking of the F-35B.
- Tilted prop rotor: A statically tilted rotor wouldn't do a ton at low angles, and at high angles would be extremely inefficient in forward flight.
- Thrust vectoring: As others have noted, this would likely only cancel out lift gains from the flaps.
- Other options: See below
- Dynamically tilting prop rotor: perhaps, But this would seem more mechanically complex to me (rotational drives, shaft bearings and CV joints, morphing ducts, keeping a healthy flow to turbine, etc.) than a seperate shaft driven fan and likely wouldn't end up saving you weight. Not worth pursuing in my mind.
- Aggressive leading edge slats: It seems that leading edge slats don't have too significant of an effect on the moment of an airfoil. I haven't seen anything listed in any NACA testing I've seen, it seems like they treat it as a given that there isn't too much of an effect. This design compendium seems to agree on pg.256 (pdf pg.277).
This is all, of course, addressing your interest in using flaps as a high-lift device. You might want to think about other high-lift devices, either in their place or to mitigate their use in your design so you don't have as much of a moment increase to deal with in whichever method (shaft-driven fan, please) you might decide on if you go that route.
The first, in my mind, would be blown airfoils and related high-lift devices. You're already looking at "ducts from the engine output," you could instead throttle the exhaust of the engine and route the exhaust to slots placed midway along the wing to increase boundary layer energy, thus increasing lift. If you want to get really funky, you could look into reverse flow slots(see 4th paragraph of "mechanism" section). The paper by Wake, Tillman, Ochs and Kearny on the matter is fascinating and worth getting your hands on. I'm worried what effect swept wings (inc. delta wings) or even the lack of an high aspect ratio would have on its effectiveness, though.
Alternatively, you could look into making the exhaust a venturi vacuum generator and play with boundary layer suction. This is apparently very attractive on highly swept wings, so it might be pertinent to your design.
The Shinmaywa US-1 and US-2 flying boats use a dedicated turboshaft engine inside the hull purely to provide high-pressure air to their boundary layer control system, and the resulting STOL capabilities are mind-bending, go find some videos on it. You wouldn't be able to get that much benefit from harnessed exhaust exhaust, as the Shinmaywa uses it in conjunction with some nuts flap configurations that would be far too aggressive for your planform, but it can give an idea as to the potential of boundary layer control.
Note: sorry for the lack of sources near the end. Limited on #links until I get more rep I guess.