I am working on a related problem, attempting to connect a guitar to DAW s/w running on an iPad Pro with USB C. My goal is to hear output from the DAW s/w running on the iPad Pro via the audio interface's monitor/headphone output. While the details of our challenges are different, the root cause may be the same.
I have more to learn about this. Having said that, here is what I've picked up about USB C, which may help you understand where your mic problem comes from, even if it doesn't solve it.
People use the term “USB C” to refer to a couple of separate, but related items. The first is the physical connector, which is a USB Type C connector. It was developed by the USB Implementers Forum, a standards body of which Apple is a member, to achieve several aims, such as to increase the data transfer speeds, increase the power which can be delivered, have a connector that’s more convenient to use, etc.
“USB C” is also used sometimes to refer to standards which define the electrical signals sent through USB connectors, which can lead to some confusion with users. A more clear (and cumbersome) way to refer to the electrical signal standard is to use the name of the standard implemented on a specific device. Examples of the electrical signal standard names are USB 3.1 Gen 1 or USB 3.1 Gen 2 or Thunderbolt 3 (or USB 3.2, or USB 2.0). All of these different electrical signal standards can be implemented on a physical USB C connector. In the case your mic, what I've read indicates the Apple-designed Lightning connector on it uses the USB 2.0 electrical signal standard.
So here's the part that’s relevant to me, and possibly also you. The USB C Connector on a specific device, such as an iPad Pro or a Mac, can be implemented to work with one or more of the electrical signal standards. And the accessory you plug in to that device may or may not have implemented the same electrical signal standard.
Here's an example: On the new MacBook Pro 16", the physical "USB C" port is set up to be a Thunderbolt 3 port, as far as the electrical signals are concerned. If an accessory is connected and does not use the Thunderbolt 3 standard, then the Mac’s OS is designed to change the electrical signal standard to USB 3.2 Gen 2 to see if that works. If there's a match with either Thunderbolt 3 or USB 3.1 Gen 2, then the signals connect and it generally works. If there's not a match with either standard, then a connection is not made, or at least a complete connection is not made. You may be able to power the external device, but no data transfer will happen, for instance.
Compare that to the USB C connector on the 3rd generation iPad Pros, introduced in Oct 2018, which use the electrical signal standard USB 3.1 Gen 2, but not Thunderbolt 3. So an external device device that uses a Thunderbolt 3 electrical standard through a USB C connector can connect to a MacBook Pro successfully, but not with an iPad Pro even though they both use the same physical USB C connector.
There's another thing going on here, too. I believe Apple’s assumptions about how a Mac should handle external devices differs from assumptions about how an iPad should handle external devices. The Mac, being a "computer", should be able to work with a very broad range of external devices without the user having to download extra drivers, configure applications, etc. The MacOS does the heavy lifting of figuring how to to make it simply work for the user.
However, iOS devices, such as the iPad Pro, appear to be designed with different assumptions, where fewer devices are connected to the iPad, and apps rather than the OS should provide the software support to connect a specific device. Also, in the past Apple has appeared to want to exercise some control over what types of devices can connect to iOS-based devices. For example, you can physically connect a USB C hard drive running the USB 3.1 Gen 2 standard to an iPad Pro with the same connector and electrical signal standard, yet the iPad will not use the hard drive. So even when the physical connection and electrical signal standards are compatible, devices can fail to connect successfully.
Not saying I agree or disagree with this approach, and I could be completely off-base; it’s just what appears to be happening.
So, when connecting an accessory to an iPad Pro, both devices have to 1) use the same physical connector and 2) use the same electrical signal standard. And 3) Apple (or possibly a 3rd party app) supports the connection to the accessory with software. And if you insert an additional device in the chain (such as a third party adapter), then there are more opportunities for signals not to connect. As I said at the beginning, this doesn’t solve your (or my) problem. Understanding the problem, however, can help find the right product to solve it.