I have to K-base sidecars, maybe thesesolutions will help.
I don't see why this type of suspensions would be useful for a sidecar still using the original motorcycle frame. Modern kneelers use them, but they get rid of the motorcycle chassis, they're designed around a central pole, with all elements attached on both sides. This solution with a front suspension using two superposed triangles moving around a longitudinal axis may be good for geometrical reasons: but it will imply to strenghen a lot teh sidecar chassis, meaning adding a lot of weight - weight you can't save elsewhere iy you still use the motorcyle frame. Furthermore, this weight is located at the front of the sidecar, very far from the center of gravity. It can be useful in case of sleepy front wheel in acceleration, but not good about handling.
Another concern is about the front wheel travel: it makes the width varying according to the suspension travel. It's OK with short travels, on road racing sidecars. It's OK on a car, because both front wheels vary. But on a road sidecar, with longer suspension travel it may cause weird results in roadholding...
The alternative is to still use a twin triangle suspension system, but moving around a transversal axis.
This will guarantee the width to remain the same, and the trail issues are not a problem anymore because, forming a parallel system, the trail remains always the same.
It also permits to attach the suspension system nearer the center of gravity, using the original frame or the subframe instead of the sidecar frame. Saves weight.
Another concern about the subframe sketch you showed. Using the central stand bolting points under the gearbox is OK, this place is very solid because it's designed to support all the weight of the bike, but the points located on the engine lower case...Uh uh! This part may be not strong enough to support the efforts, and any crack will instantly drain the oil. You should better use the engine to frame junction to attach the subframe, they are much more solid. Furthermore, the tubes passing under the engine oil case will lift the whole sidecar for ground clearance reasons - and you don't wan't to have a high sidecar for roadholding reasons. A subframe around the engine is better than a subframe under the engine.
I have to K-based sidecars:
This one uses a Side Bike front end, designed for the Yamaha XJS900. It also uses a sidecar steering wheel, but this doesn't matter here.
The front end is very light and simple. Note the subframe design: the upper fixation is bolted on a plate welded near the fork axle: it's the only attachment that needs welding.
Then a tube runs around the engine both side, providing fixation to the front end, and re-using the engine to frame points.
Note that this solution implies to re-route the exhaust system, under the engine, that implies to have a high motorcycle to preserve ground clearance. I did not designed it, but if I had, I'll chose another option, with lateral exhaust, to lower the bike.
Handling and maneuvrability are excellent, but I can't tell you if it is because of that front end design or because of the steering sidecar wheel.
My other sidecar is a K1 using a double triangle suspension system:
Also a french design, made by BCS, Benoist Constructions Spéciales.
It uses 2 triangles, moving around two transversal axis, attached on a module that is not fully integrated to the bike nor sidecar frame. This because it was an aftermarket modification, the sidecar was originally designed with an Earles fork, but it was not satisfying and was modified.
The steering system is complex, quite hard to turn! But the stability this system provides is extraordinary. May the road be bumpy, it does not matter, the trajectory the sidecar follows remains absolutely unchanged. You can ride without hands on the bars, it won't move.
The front wheel is prone to understeer when accelerating frankly (franticly!) on wet roads. I think this is due to a lack of weight on the front end, but also to the fact that the front wheel is very far away trom the bike (because of the aftermarket mod, the front module is not well integrated). But I have to test a new tyre before concluding, this one is old and not very sticky anymore.
I'm afraid I have no pictures about the subframe design on the K1.