I also think splitting off the Dante to FOH is the best solution. You can send Dante channels to more than one device at a time, so your artist on stage could be manipulating their Dante feeds while you record the same channels at FOH.
If you aren’t running a redundant connection between the mixrack and the artist, you can simply use the other Dante plug in the I/O card for this purpose. If you are running redundant or have otherwise maxed out those connections, a simple “dumb” 4 port network switch will work fine. I currently use several old consumer grade routers for this very purpose. Of course you need to turn off the DHCP service and Wifi, but otherwise they work great to connect a few Dante devices on a “Dante Only” network arrangement.