I have a wireless 802.11ac router, into which an external hard drive holds my entire ~2TB music collection.
As Jim said, this is almost definitely the weak link in the chain. External drives attached to routers are notoriously slow, particularly if they are a USB drive, where the connection is usually USB2 rather than USB3. Performance of these types of connections is typically well below the maximum speed even of USB2. Priority on the router is given to network traffic, not serving files from the hard drive. If the external drive is attached by Ethernet to the router, it would perform much better, if it is in a good enclosure. If there is a virus scanning solution on the router, and it is a USB drive, then the performance would be miserable.
Updating tags from a library, or the library from tags, is going to touch every file on the drive, multiple times. This will saturate the connection, and I would not be surprised if errors started to creep in. While the method described above is the correct one for this installation, it isn't going to work.
All of the above is also why you get some skips listening to SACDs and high bitrate files, even without implementing a MC Client/Server installation. A USB attached drive on a router is only really good for low usage, low performance uses, such as saving documents to a separate location for backup (not using backup software), or perhaps jpeg images. Small files, small bitrates, low demand and throughput.
One solution would be to back up the library on the Lab computer and restore it to the Office computer. I order to make this work;
1. The router attached drive must be mapped to the same drive letter on each computer, or the same network address for the drive must be used.
2. Each installation of MC must be identical, on the same drive letters, with the same settings. i.e. A standard installation on the C: drive.
3. You may need to be using the same User ID on each PC for it to work. Certainly if you are using any of MCs user capabilities.
4. There may be other issues with things like Cover Art and caches on local storage on each PC.
5. You will need to do all maintenance on one PC, I assume the Lab PC, while the other installation becomes just a clone for playback. Whenever you do maintenance on the Lab PC, you would need to back up the database and restore it again on the Office PC. Annoying.
6. There may be issues with playing music on both PCs at the same time. But you would have those issues now with your current separate libraries and shared music drive.
Basically, that solution may not work. If you try it, back up both your Lab and Office PCs, and keep those backups safe, in case you need to revert.
Another solution would be to move the music drive off the router, particularly if it is USB attached, and put it into a proper Network Attached Storage enclosure, connected via Ethernet to the router. That will give you some performance increase, but maybe not enough, unless you also . . .
Connect both PCs via Ethernet to the router, if at all possible. Especially if your router and PCs have gigabit Ethernet ports. However, if you can do this, then a MC Client/Server installation using Media Server would probably work fine anyway, and you wouldn't need to run two separate libraries, or get into "updating tags from library" and "updating library from tags", so no big performance hit on the drive. Just progressive maintenance load, which is much lower.
If you can't get the PCs on to a wired Ethernet network, and don't want to try a NAS (because it still may not be good enough with a wireless network) then you could try installing the music drive into the Lab PC, then make that PC the MC Server, and the Office PC a MC Client. In that configuration, with almost all maintenance done on the Lab PC (subject to normal restrictions of MC in a Client/Server installation i.e. you can do tagging on a Client PC, but not run Rename, Move & Copy commands), the drive connection and wireless network would probably be fast enough to serve your music without skips, and all your worries about a managing and maintaining a duplicate library will be over.
Oh, one last thing. Wake On LAN is a pain, but it can be made to work, particularly within the one LAN environment such as you have. That is a solvable problem. Also, if you are playing music on the Lab PC, the Office PC doesn't need to be on. It is only when playing music on the Office PC that the Lab PC also needs to be on, and that is the direction that WOL would need to work.