So it seems to me that with a computer (best you can afford) and any DAW software (best you can afford – some are free, but . . .), you should be able to purchase and install Dante Virtual Soundcard. It only costs around $30 for one license.
You can mix up to 64 channels in your DAW (as many as it can handle), use the virtual soundcard as your input into your DAW and then route the master back out to your live stream. (I could be crazy, but it might work!) So that computer simply becomes another device on your Dante network.
I am certainly not an expert in Dante or the iLive, but I use both and I think this should work.