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128GB drives are exFAT compatible so this is the File system you should use. If it is still in FAT32 reformat it as exFAT. 

What version of iPad do you have? 

What iOS version are you using?

What SanDisk connect app version do you have?

What FW version os on your Wireless Stick?

“Also there is often a message top left in the app screen saying drive is in USB mode.”

If you are not connected to the drive it should give you a “Not connected” page with instructions to connect or charge the wireless stick. Can you post a screen shot of this error? If the controller is not making the switch from MSC to wifi the drive may be defective.