How can I get WBPP/FBPP to write to a locally-attached NAS?

At the moment it is not possible. PI does not like NAS or online synchronized backup disks, as you have seen, if you try to do it, a series of errors appear.

PI likes local disks, no matter if they are internal or external.

You could certainly map the NAS as a network drive…

Let me play devils advocate and ask you…  WHY? 

I don't care how good your network is, from a performance perspective, running WBPP to a NAS would SUCK from a performance perspective…

You'd be much better to set up a schedule task or a script that mirrors data from a local disk where you perform WBPP, to the NAS… And ultimately, clearing the directory on the local disk too… 

Its not even that PI prefers local disks, its that read/write speed is SUPER important in WBPP… That's half the reason why FBPP was created - to reduce the IO bottleneck by keeping everything in memory.

A really good, low latency/high efficiency NAS will have something around 110MB/sec. A USB SSD will double that for performance to around 200MB/sec. a local spinning disk will double that and then some again at 550MB/sec, and a top tier NVMe SSD might be as high as 15,000MB/s. 

I would honestly avoid using NAS for WBPP at all costs if I were you.

I messed with this and Alex is 100% correct. Trying to run WBPP on a NAS is frustrating at best and disastrous at worst. There is no upside. PI won’t even run well on a drive that syncs to a NAS in my experience. I use my NAS as my long term repository of lights and masters.

The big risk you run with any DFS/RFS (Distributed/Replicating File System) is data corruption if things get touched/updated while a replication is occuring, but yes, if your sync to the NAS is done by 'watching' the target file system, then triggering a sync with the NAS when things are changed (essentially, a live, differential backup) that will incur a fairly heavy performance hit. 

What I was suggesting was having a local 'working' directory on the machine, then having a scheduled task that backs up that directory/disk (or even better, things that changed) at say, 4am.

I would never recommend any process of cloning files while they are in use as the performance hit is significant, and the risk of corruption is absolutely an issue to be considered as well.