I always had a G27 and it always worked great with Wreckfest. A possible problem really is that every single wheel is completely different to communicate with.
I have talked to an rFactor 2 dev about that problem. They have to write code specifically for each wheel, for example Logitech wheels rely on the Logitech profiler to run in the background or else the game can only receive the most basic input from Windows without sending anything important to the wheel, like proper FFB or wheel settings or something.
Other wheels turn out to be trickier and problems like this are quite common among all wheels and all games that support them. Often it's hard to find the cause and very often the cause is the driver itself.
The only thing you could try would be trying a different driver and/or windows (I'm running 7 which is probably the most stable one when it comes to that) or hope for Bugbear to find a workaround. Maybe the wheel simply takes a while to initialize in some cases. In that case a possible workaround would be to do that while loading the track instead of waiting for the race to start.
Often games try to reset the wheel everytime you try to drive to make sure the wheel's settings are all correct, but some wheels just need a few seconds to properly do that. While others do it instantly.
This is my best guess and since there will be monthly updates now, if there's actually code for that in the game, I'd suggest trying to temporarily disable it for the next build so that you can find out if that fixes the problem.
Literally the only thing that confuses me about this whole situation is the fact that your wheel doens't do this all the time, but only online when there are other cars running. Could be a bug in the game even, but that would still be linked to that specific part of the code that controls the wheel because other wheels don't have that problem