If the headset doesn't have a 3.5mm audio connection, but instead has a Lightning plug, you'll not be able to use the headset in the manner that you describe with your iPad's USB-C connection.
Lightning and USB-C are completely different electronic interfaces. Both the physical and electrical presentations are entirely different. You can't use a simple plug adapter to convert between the interface types. To the very best of my knowledge, the type of "adapter" that you seek doesn't exist - either from Apple or third-party manufacturers.
A couple of years ago Apple did introduce an active USB-C to Lightning Adapter - specifically designed to connect Lightning accessories, such as your Lightning headset, to USB-C host devices. This Adapter incorporated an active interface converter - but did not offer simultaneous connection to a power source with which to charge the host device.
USB-C to Lightning Adapter - Apple (UK)

Apple's website suggests that this Adapter is no longer available directly from Apple. This Adapter, should it be of interest, might still be available from an authorised or independent Apple retailer while stocks last.
For avoidance of doubt, this Adapter supports connection of Lighting accessories to an iPad/iPhone's USB-C port. This Adapter does not have a separate charging connection.
Given the cost of the Apple USB-C to Lightning Adapter, assuming that you can even find one to purchase, it may be an easier and much more cost effective to simply replace your existing headset for a USB-C equivalent.
Alternatively, an altogether better solution might be to consider a wireless (Bluetooth) headset or earbuds. Apple's AirPods are an excellent choice; many third-party alternatives are also widely available.