How do I insert LB's IP with an irule?

To clarify...

HTTP_REQUEST_SEND is in the server side context as it's when the BIG-IP is sending the request to the server. So if you use [IP::local_addr] in HTTP_REQUEST_SEND it should give you the source IP the BIG-IP will use.

In the last example you posted, you're using [IP::local_addr], but specifying the clientside context. So that should equate to the VIP address (the address local to the VIP on the clientside connection).

So you should be able to use this:

...
when HTTP_REQUEST_SEND {
   set bigip_source_to_node [IP::local_addr]
...
   clientside {
      HTTP::header insert "Source_to_node" $bigip_source_to_node
...
   }
}