I now have a support case opened and it appears that Harmony does have support for ATWINC3400-MR210CA:
"If MPLAB is used as development environment, then harmony is the recommended framework. Harmony also contains example applications for winc3400."
Also, FreeRTOS is not a requirement:
"No, there is no need of using FreeRTOS. WINC driver along with SPI peripheral is enough to communicate between MCU and WINC."
Unfortunately, the Harmony support is for SAM (Atmel?) devices, so you have to translate to PIC32:
"The default applications present in the Harmony Framework is made for SAM devices that have to be ported to PIC devices in your case."
Now I just need to find out if the latest Harmony will work with MPLABX 5.25, which I'm currently using, or if I will need to upgrade MPLABX as well.