irule to insert a client cert for authorisation to a website

From what I see your requirement, you dont want to have mutual authentication for your clients. But the external vendor website which is your pool member, requires to have cert produced to access it.

You can simply configure a cert in your custom serverssl profile and pass it. By default the cert is none.

In case the vendor would accept only cert CN's, have that installed on LTM and map it to the serverssl profile. This way your clients can connect to the VS without any cert and on the backend LTM will be providing the cert while connecting to the external website.