I think I meet the "necessary" test since the guide mentions that it has to do with the fact that a trip is required and doesn't mention anything regarding class of travel to meet that requirement.
Additionally, the guidance is clear that HMRC wouldn't consider an expense unreasonable just because a cheaper alternative is available. The guide goes as far as giving an example of an employee who gets reimbursed by the employer for standard class travel and buys a First class ticket, making it clear that the difference is eligible for tax relief.
Also, other people, including c-w-s said in the thread quoted above that they successfully done this.
I've resubmitted my self-assessment for last year as I had those receipts available and will update this thread when I hear more.