Yet more fundamental misunderstanding of what an ATPL skills test is.
A raw data ILS is a feature of any skills test, including for type rating issue, it is bread and butter and as stated shouldn't be an issue.
An ATPL skills test is far more a test of non technical skill to the required level to command a multi pilot aircraft. See Part-FCL Appendix 9 - 15
15. The following matters shall be specifically checked by the examiner for applicants for the ATPL or a type rating for multi-pilot aircraft or for multi-pilot operations in a single-pilot aeroplane extending to the duties of a PIC, irrespective of whether the applicants act as PF or PM:
(a) managing crew cooperation;
(b) maintaining a general survey of the aircraft operation by appropriate supervision; and
(c) setting priorities and making decisions in accordance with safety aspects and relevant rules and regulations appropriate to the operational situation, including emergencies.
Whilst within a supportive airline environment with robust SOPS, you may well expect to pass such an assessment, as a third party candidate with some random off the shelf examiner it will always be an unknown. My personal pass rate is in the region of 90-95%, candidates definitely fail by not taking the skills test for what is is and treating it as a recurrent LPC which is a very different beast. With a recurrent LPC there would be training between attempt 1 and 2, this would not be acceptable on a skills test.
OP is definitely right not to go for an ATPL skills test with a third party examiner, as if they do fail, even for a poorly conducted test on which they win an appeal, the affect on their rating affects their livelihood. Better to wait 6 months and do it in house with the support of your airline. Remember to complete the notification of skills test, this is not a recurrent check of ongoing proficiency, but the initial test, and a demonstration of skill.