It sounds like you need simple Write-Once-Read-Many-times (WORM) media. The most common media type that fits this description would be DVDs. If you trust that the DVD-R discs are actually blank (which will be the case if you get them from any reputable source), then you can write the video to disc, eject the disc, and never have to worry about it again. DVD-R discs are cheap enough that you never need to worry about reusing them (and for anyone who is worried about the environment, there are many craft projects that can be done to recycle old discs). For that matter, you could even use a Blu-Ray drive to get even more storage per disc, though recordable Blu-Ray discs are more expensive.
If you don't have an integrated optical drive, USB-connected optical drives are very cheap these days, and quite reliable.
All you need to do to make sure you stay safe is to make sure that you only put in blank discs and that once a disc has left your possession you never put it back in the system.
Of course, there are some downsides to this approach. First, you might end up using an awful lot of discs, depending on how much you need to export. Second, disc burning is slower than the transfer to flash-based media as is copying the files off the disc, though direct playback speed on other devices shouldn't be a problem - after all, that is what DVDs were created for. Finally, many modern computers don't have optical drives anymore.
That said, if you want to ensure that you don't get infected by malware from flash media, this sounds to me to be the easiest way to do that.
One other important note is that if you don't completely fill the disc, many programs will let another user add additional content afterwards, unless you explicitly "Finalize" the disc. Even so, I think that this is outside your threat model, as it will only be an issue for someone who is deliberately writing the malware onto the disc. This is because, unlike a USB flash drive, writing to optical media cannot be done just by writing to a filesystem; it takes actually telling the burning software to write to the disc.