Have there been any attempts to measure profiles through deep convective systems using dropsondes?
NOAA flies routine dropsonde missions into tropical convection if there is anticipated threat to land.  You can find information about these missions in the routine NHC updates for each storm.  This data is used in estimation of hurricane intensity and some of the models assimilate this data to improve storm track and intensity forecasts.

You can learn more about NOAA's hurricane hunters here.



Dropsonde missions over the continental US are problematic.  The mesoscale predictability experiment (MPEX) use dropsondes to determine if better upper air observations could improve predictability in the mesoscale.  One big issue they had is restrictions in where they could drop from the FAA.  They had to provide notice well in advance of the flights and could only conduct missions during certain time windows.  This provided big restrictions in dropsonde missions and would probably make it unsuitable for drops into deep convection (though one might argue with air traffic routing around such convection, that would make drops feasible from an FAA standpoint).

There were some recent talks about MPEX at the 27th Conference on Severe Local Storms in a session about MPEX.  In particular, a talk by Morris Weisman (video) talks a bit about the dropsonde missions.



From another perspective was the DC3 project, which I provided nowcasting in support of aircraft operations.  This project didn't use dropsondes, but instead used aircraft in vertical flight profiles to collect observations from near-surface to tropopause levels.  These profiles were in the vicinity of deep convection, but not within.  This project was primarily interested in how the chemical composition of near-surface air was modified by convection and then transported in upper air outflow.  

These aircraft were not dropping dropsondes but I'm not confident a UAV would have been able to fly too much closer to convection to drop into the storms.



In summary, maritime dropsondes are common for getting in-situ measurements of tropical convection, but for continental applications FAA restrictions on dropping things from airplanes makes routine dropsonde missions unfeasible (and I'm assuming these would be done by UAVs as getting NSF aircraft assets to do this would be restricted to short term experiments and not routine operational missions).