It seems that the ansible lookup plugin does not adhere to privilege escalation and it is not clear to me if this is by design.
I have looked for an answer to this, but though I have found many similar questions, I haven't yet seen one which seems to answers why the following playbook behaves like it does.
---
- hosts: localhost
  become: 1
  tasks:
  - name: cat file
    command: cat /home/bob/.ssh/id_rsa.pub
    register: cat
  - debug:
      msg: |
        dog: {{ cat.stdout }}
  - name: add the variable
    set_fact:
      rsa_key: "{{ lookup('file', '/home/bob/.ssh/id_rsa.pub') }}"
    delegate_to: localhost
The result of running this play is that the command module "works" while the lookup module does not:
PLAY [localhost] *************************************************************************************************************************************************************************************************************************************************************
TASK [Gathering Facts] *******************************************************************************************************************************************************************************************************************************************************
ok: [localhost]
TASK [cat file] **************************************************************************************************************************************************************************************************************************************************************
changed: [localhost]
TASK [debug] *****************************************************************************************************************************************************************************************************************************************************************
ok: [localhost] => {
    "msg": "dog: ssh-rsa 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 ansible-generated on rh1.local.home\n"
}
TASK [add the variable] ******************************************************************************************************************************************************************************************************************************************************
[WARNING]: Unable to find '/home/bob/.ssh/id_rsa.pub' in expected paths (use -vvvvv to see paths)
fatal: [localhost]: FAILED! => {"msg": "An unhandled exception occurred while running the lookup plugin 'file'. Error was a <class 'ansible.errors.AnsibleError'>, original message: could not locate file in lookup: /home/bob/.ssh/id_rsa.pub"}
PLAY RECAP *******************************************************************************************************************************************************************************************************************************************************************
localhost                  : ok=3    changed=1    unreachable=0    failed=1    skipped=0    rescued=0    ignored=0   
I'm running this under a user, which is NOT "bob", and with become_user = root and become_method = sudo. Any ideas or a confirmation that indeed the lookup plugin ignores privilege escalation statements, would be appreciated.