You are running a non-MSI driver install EXE from within your MSI? Correct? Or maybe it is an MSI wrapped in an EXE?
Do you have Installshield Premier? Could you use a suite project and install the EXE via the bootstrapper before (or after) the MSI install? I have honestly never used this feature, but running setups in sequence is what it is for. Embedded custom actions in MSI files kicking off EXE files are notoriously unreliable. This is - in my opinion - especially true if you are running with managed code as well (which I think you are).
In the long run managed code may yield safer custom action code (security-wise based on CAS), but for now it seems to cause unwanted runtime dependencies - especially for very large-scale distribution (global distribution) targeting diverse Windows versions (Vista, 7, 8, 10).
I am told it takes a while to get used to Installshield's suite feature, but maybe it is better for you? You can run EXE files, MSI files, patches and zips in sequence. Some fiddling to define uninstall and upgrade behavior I guess and lots of testing. I am pretty sure corporate application packagers would be happy to see a suite rather than an MSI with lots of strange stuff embedded in it.
UPDATE: Once you have compiled a suite setup.exe file it can be extracted as described here: Regarding silent installation using Setup.exe generated using Installshield 2013 (.issuite) project file
Alternatively you could try to extract the setup.exe files for the driver setup and install the drivers as regular MSI components and run DPinst.exe to install / uninstall the drivers (tool from DIFx). Also quite clunky - especially when you need to include uninstall.
Your driver setup likely uses DPInst.exe already. I would check if you can extract an MSI from the EXE and use it instead of the EXE to include in the suite project. Some hints for how to deal with setup.exe files (extraction, runtime paramenters etc...): Extract MSI from EXE.
WiX has the Driver element in one of its extensions to deal with driver installs. I have never had the chance to test it.