Charging Garmin Instinct while recording an activity

This is a great question, and no reason it couldnt be done but placement of the charging port is not ideal and would need a very slimline "ribbon" connection.

but not sure of their compatibility with the Instinct.

You can check the compatibility of each product by clicking the tab Compatible Devices

I may not have checked all, but did not see any clip for Instinct in Garmin's store. In contrary, you can find some L-shaped charging adapters for Instinct at 3rd party vendors like AliExpress, Amazon, eBay, etc -

An example lookup at AliExpress: garmin instinct usb | Aliexpress

Yeah seems the design choices for the port location would make both solution not workable for instinct 2. As you can see the port it aligned against the area where the strap is attached. 

No market for a solution so would need a custom cable made which would be tricky as depth would need to be very low still.

I did not test it, but this one could perhaps work:

... and the puck-shaped adapter could work too - you'd probably need a strap extender, so that it can fit under the watch, and a HRM chest belt, since the optical HRM would not work through the puck.