I've seen a few posts with successful nice!nano wireless builds
Is there room for a battery under the microcontroller?
I think so. The sockets included with the kit are 4.2mm high. I believe a common battery is the 301230 which is 3mm high. The microcontroller should be installed upside down like the ProMicro I believe so will lose some space to components on the nice!nano. This reddit post recommends more than 3.5mm.
Would to love hear of your experience so I can update this with better guidance.