What are the business reasons? Unless they are denying all other relevant employees AL at that time for the same reasons, you're being treated differently and you don't have to put up with that.
Have you checked your contract? I've only ever heard of AL during a notice period being 'frowned upon', never seen a contractual ban on it. In fact the only time I can recall seeing it mentioned contractually is when the company makes you take any accrued AL instead of paying it out.
In terms of your legal position - technically they could sue you if you just walked out tomorrow and it caused a detriment to the company, however this is extremely unlikely. For example, if you were due to deliver a project and as a direct result of you not doing so, the company lost a client or suffered financial penalties. They would absolutely not take legal action against you for taking one day off.
They could put you through disciplinary/sack you if they deem it gross misconduct. Obviously you're under no obligation to engage with a disciplinary process, but it could be mentioned on your reference. Again though, it's very unlikely as it's a waste of time and effort on their part.
Do you even need a reference if your job is secure?
My recommendation (once you've checked your contract) would be to tell - not ask - them that you will be taking that day off. Tell them you're not aware of anything in your contract preventing you from taking AL during your notice. Tell them that any business need must be applied fairly and equally to all relevant staff, and you will not be treated any differently than your colleagues, or bullied, just because you have handed your notice in.