Irule direct traffic to external site

I think that this is how the F5 is sending the external traffic, and this gives me an error.

curl --header 'Host: www.externalsite.com' https://192.168.1.1/merchant.jsp

What kind of error do you receive?

I guess you have to provide hostname not only in host header but in TLS SNI field too. SNI extention allows server to show proper certificate depends on target hostname.

Here you can find the IRULE to inject SNI into server-side connection https://support.f5.com/csp/article/K41600007