As the title suggests, is it possible to continuously charge the X-T200 while filming? I haven't figured out a way to do this, plugging in while shooting doesn't seem to do anything.
You likely want a DC-coupler. I don't see any listed for the X-T200 (which is a fairly recent camera), but the DC coupler acts as a dummy battery, so if your camera uses the same battery as a previous model you can use a DC coupler listed for the previous model.
The USB port on the vast majority of interchangeable lens digital cameras is there for communication purposes that does not include the ability to charge the camera's battery.
Though this is slowly beginning to change in some recent models, such cameras are still in the minority of models currently on the market.
Your Fuji X-T200 is one of the few models that does allow battery charging via the USB port when the camera is turned off. The online Fuji X-T200 Instruction Manual makes clear that the camera must be turned off to use the USB port for charging the battery.
One reason for this restriction is that the camera draws two amps (2A) minimum when turned on and can draw up to four amps (4A) when operating. Most USB based power sources provide far less than that (typically anywhere from 0.5A to 1.5A). Even if you have a large power bank with a very high total capacity, it usually can only supply power at less than your 2A minimum requirement. In other words it would take your power bank longer than an hour to supply the camera with the amount of power it consumes in an hour. Think of it like a water faucet. It doesn't matter how big your storage tank is, the amount of water you can get per minute is determined by how much you can open the valve in the spigot.
Your Fuji X-T200 is not one of the very rare ILCs to have the ability to charge a battery inserted in the camera while the camera is turned on. If you need to leave the camera on continuously for long periods of time that exceed the capacity of a single battery, then follow the advice in xenoid's answer and look for a DC coupler, also known as a "dummy battery", that fits cameras that use the Fuji NP-W126S battery.
Notice that this dummy battery allows it to be plugged into two separate power banks, or other sources of USB power, in order to provide the needed current of 2A in real time.