HooToo + VLC Player
HooToo makes travel wifi routers that support a media center network share. You just need to plug your USB stick into the travel router, turn it on (be sure media sharing is enabled), and connect the the travel routers wifi network.
VLC Player can then use that share to play videos on the USB device. It should find it automatically. In fact, it can stream or download the videos on your flash drive.
Yes, this is not a "just works" case, but it works for me.
(Alternatively, you can use Readdle's Documents, which can put files on the flash drive, but connecting to HooToo is a bit tricky and it supports fewer media formats.)
So why HooToo? It's just a decent brand. I own one that replaced my wall charger and battery pack. Alternatively, I have a 32GB flash drive on my home router. Both routers have SMB/Samba shares and media shares built-in.
(All that aside, rumor has it that camera adapter SD or USB dongles can be used to transfer MP4 movies in, but not out. I also hear rumors that the new TV app has trouble with ripped media. It sounds like a gamble to me, but is worth looking into.)
Fun Fact: VLC can act as a wireless media device to stream and download movies from, while on the same network. I use this to put files on my older iPad from my newer iPhone which has more space. My phone often replaces my flash drive.