First attempt: Changing the routing of the IP address
In the internet the smallest block which can be announced is /24 for IPv4 and /48 for IPv6.
This means that two IPv4 addresses which have the same 3 first bytes must be routed to the same ISP.
Example: The addresses 192.0.2.43 and 192.0.2.65 both start with 192.0.2, so they must both be routed through the same ISP.
Routing 192.0.2.43 over another ISP would only be possible if all customers using an address in the range 192.0.2.xxx would switch the ISP at the same time.
... and even then I'm not sure if this would really work.
Second attempt: Changing the local IP address of the connection
(In this case you have two different IP addresses for two different ISPs but the chat program you are using is capable to switch between the two IP addresses without dropping the connection.)
At least there are experimental protocols which allow to do so; maybe there are also protocols suitable for commercial use which can do this.
However such protocols would probably use UDP and work on OSI level 5 or above. This would mean that such protocols are not in the scope of this website any more.
Using such protocols would however definitely require special software both on the customer's PC and on your server.