A switch will update its MAC address table from any frame it receives on a port. Switches aren't interested in IP, or any other layer-3, addresses, only layer-2 (MAC) addresses. ARP is a host protocol to resolve a layer-3 address to a layer-2 address.
Since the layer-2 addresses of the PCs aren't changing, the switches are unaffected by the change in IP address.
You can clear the ARP cache on the router. It will eventually time out anyway, probably during the time it takes you to notice that PC A is down and move the IP address to PC B.
You are trying to arping you own address, and you don't get a reply. If you arping the gateway address, you should get a reply.
First, a layer-3 switch is primarily a layer-2 switch. The layer-3 router of the switch is separate from the layer-2 switch part, connected to the layer-2 switch via (a) virtual interface(s), as if it was a separate external device. The router is only concerned with what are connected to its layer-3 interfaces, virtual, or physical.
Just as if it was a separate router, the router of the layer-3 switch maintains an ARP cache for a connection to a layer-2 network, but it only knows which layer-3 interface to which it should send traffic, not which particular layer-2 port to which an individual host is connected.
The layer-2 switch part is just like any other layer-2 switch, it maintains a MAC address table which tells it to which port a MAC address is connected, and it knows nothing about layer-3 addresses.
The router ARP cache and the MAC address table entries are temporal; they time out after a certain period of time. This time period may be configurable, depending on the switch model. The time period is probably much less than the amount of time it takes for you to notice that PC A is down, plus the amount of time it takes for you to configure PC B with the IP address from PC A.
The layer-2 part of the switch is the part concerned with the physical switch port to which PC B is connected, and it only cares about PC B's MAC address, which I doubt you changed when changing its IP address.
The router's ARP cache is only concerned with which MAC address should be related to the IP address, not the physical switch port to which PC A or B is connected. The ARP cache entry may be stale, but not for long, and probably not at all.
Hosts, including routers, are constantly timing out ARP cache entries when traffic isn't received from the IP addresses. Assigning the IP address to PC B will cause it to ARP for the router's MAC address when it needs to contact the router, and that will change the router's ARP cache entry for that IP address to point to PC B's MAC address.
In short, there is nothing you need to do to the switches when PC A is disabled, and you assign the address from PC A to PC B.