I'm the former maintainer of the Crashlytics SDK for iOS/macOS. I'm relatively unfamiliar with the Android version of the SDK, and definitely unfamiliar with Android in general. But, I'll give this a shot.
What you want to do is something that has been requested on the iOS side a few times. I would have loved to do it actually, because it seems pretty terrible to force end-users to incur these costs. However, the iOS SDK's networking and start up routine are both very complex and very delicate. It is highly challenging to guarantee that crashes are delivered and that there are zero possibilities for inconsistent states. I believe that Android is simpler here, but I cannot say this with authority.
The iOS SDK, however, does have some hooks for additional client-level functionality. Check out the warning around one of those APIs:
* @warning Just implementing this delegate method will disable all forms of synchronous report submission. This can
* impact the reliability of reporting crashes very early in application launch.
Basically, in order to satisfy the contract of this particular API, some techniques to improve reporting reliability have to be disabled. The thing is, sometimes it's worth it. Lots of apps decide to make this tradeoff. Many apps also delay initializing Crashlytics to eek out extra performance. This has a huge impact on reporting reliability, but that's another tradeoff app developers have to make.
I think you should seriously consider just not enabling Crashlytics in these situations, if you can easily detect them. Maybe Android even allows end-users to do this on a per-app basis? In that case, you'd never get any reports anyways. I would imagine that your user base is diverse enough that missing some reports in these situations wouldn't be that terrible. Or, perhaps you'd like to surface it as a user-facing option.
You could even do something totally crazy, like override Thread.setUncaughtExceptionHandler
yourself, and buffer up exceptions during this situation to disk. And then, replay them to Crashlytics when things are better. Turn it into an open source lib. I bet people will love it! Possibly not the Crashlytics' Android team though ;) (Hi!)
This is also basically the same recommendation Gastón offered above, with just some extra context around what I've seen on the iOS side. Also shoot the Crashlytics people an email asking for this. I think it's a great idea.