I've read about how ionospheric discharges like jets can be initiated/triggered by lightning, and how lightning can propagate upwards from clouds. What I was wondering is if this can happen in reverse. In other words, can ionospheric plasma disturbances/discharges effect/trigger troposphere to ground lightning?
Apparently it is possible for the ionosphere to influence the conditions in the lower atmosphere.
As it turns out, at least in the arctic regions, the ionosphere can have quite an impact on the temperatures.
Geomagnetic storms provide large numbers of energetic electrons, which follow the magnetic field lines. As the magnetic field lines only intersect with the atmosphere in the polar regions, those electrons will preferentially precipicate there. What follows is Ozone destruction and generation of $NO_x$-species in the Mesosphere and Stratosphere. $NO_x$ molecules are classical long-lived Ozone killers.
The Ozone depletion leads to direct heating of the Polar surface of up to $\Delta T \approx 4.5 K$ by increased UV irradiation. Furthermore, the $NO_x$ can be transported towards lower latitudes, and geostorm induced chemistry seems to have influenced mid-latitude Ozone concentrations measurable in the past for short times. A nice summary also in the Science case for the EISCAT3D-radar (not paywalled, and one can skip directly to page 7 where those effects are described and further referenced).
Those indirect effects are intriguing in that, while some might argue that the ionosphere cannot have any influence on the troposphere, it is forgotten that the radiative energy balance links the whole atmosphere. And it is this balance that can be altered by geostorm induced ozone destruction in the middle atmosphere. It's not exactly what OP was going for (no rise in lightning strikes in the troposphere), but the ionosphere knows their own version of lightning , called Sprites (also a more recent finding in research), which are sure to appear with those events.
Keep in mind that this is still ongoing research.
While a correlation of geomagnetic storms with changing polar temperatures has been already noted in the past, and seems undisputed, the exact mechanism responsible for the correlation is still under scrutiny. Personally I feel there must be more to it, as $4K$ temperature difference just from increased UV irradiation seems a bit much.
The ionosphere is too high up to affect the weather far below in the troposphere. It's more the other way around. Lightning discharges in the troposphere sometimes send vertical columns called sprites and jets up into the ionosphere. The sprites are coloured orangey red or sometimes greenish blue, the same colours as aurorae, and for the same reason. One colour is from the excitation of oxygen ions, and the other comes from nitrogen ions, but I forget which colour comes from which gas. I think red comes from nitrogen. Jets are more white in colour, longer, straighter and less variable in shape than sprites. Both phenomena are comparatively new discoveries, being first photographed in 1989.
The ionosphere begins about 35 miles up, some 18 miles above where the ozone layer leaves off. It is chiefly notable for its ability to reflect radio transmissions, enabling them to follow the curvature of the Earth though in other circumstances they would travel in a straight line. The layer which does this is called the Heaviside layer. Meteors begin to burn up in the upper ionosphere.