I actually managed to get things working the way I need them to work.
I'm not really a computer science guy, so my terminology on environments etc isn't very precise, but I'll give it a shot to describe what's happening.
I've set the virtualenv parameter in the Web Apps tab to direct to /home/timmonspatrick/miniconda3, which is my miniconda installation directory. I've installed RDKit and other modules with conda, which makes them available to Anaconda python interpreter. I haven't changed the PATH settings; if I unset the virtualenv parameter and revert to using the system python, the conda installed modules are not found.
The problem lay with getting the C libraries to work. The system version of libstdc++.so.6 was a simlink which pointed to libstdc++.so.6.0.19, which does not provide CXXABI_1.3.11, which is required by libRDKitDataStructs.so.1.
The miniconda installation provides its own libstdc++.so.6, which simlinks to its own libstdc++.so.6.0.26, but the system one located in /usr/lib/x86_64-linux-gnu is discovered ahead of the one in the miniconda installation.
I downloaded libRDKitDataStructs.so.1, (well actually the file that this simlinked to), opened it with Notepad++, replaced all references to libstdc++.so.6 with references to libstdd++.so.6 (important to retain the same number of bytes in the file, hence just changed one letter), and created libstdd++.so.6 as a simlink inside the miniconda3 directory to the miniconda provided libstdc++.so.6.0.26. Because the system itself does not provide a file named libstdd++, the one I created in the miniconda directory gets discovered instead, and links to the anaconda version of libstdc++, and the code works. (Well, technically I had to replace the libstdc++ references in 50 other .so files, but then it worked).
I'm pretty sure that the python interpreter is still the system one, as the output of sys.version gives a different result than the anaconda python invoked from Bash, but as long as the conda packages work I'm happy.
Thanks for your help @glenn and @giles, really appreciated it :)