Link Aggregation causes all volumes to disappear

Truthfully, bonding is generally not worth the trouble on a home network.  The RN104 isn't fast enough to keep up with a single gigabit connection, so there really isn't much (if any) performance gain. 

Even with a faster NAS, LACP still limits each data flow to 1 gigabit.  You'll sometimes see a throughput increase when multiple devices are accessing the NAS simultaneously, but even then it depends on how the router and NAS map data flows to the two NICs.  That's done with a hash, so there is no guarantee that both NICs will have a balanced load in both directions.  With 2 simultaneous clients, there's only a 1 in 4 chance that both NICs will be used in both directions.  

That said, it should of course work.

1. When you set up the bond on the NAS, did you also select LACP?
2. Did you try disabling the bond, and then check both connections?  That is, connect NIC 1 on the NAS to port 3, then disconnect it, and connect NIC 2 on the NAS to port 4.  Confirm that both connections work.
3. With the bond enabled on both devices, can you access the NAS web ui from the PC?
4. If you are using jumbo frames, try reverting back to the standard MTU. 

Can you clarify what you mean when you say the shares are disappearing?  Are you getting an error when you try to access them with file explorer?  Or are you seeing something else?

BTW, the hash type isn't negotiated.  Its a sender policy only, and it often is different in the NAS and the switch/router.  Looking at the manual, your router doesn't let you set the policy (and probably is using a layer-2 hash).  That is often the case with Netgear switches and routers.