Sorry, but what's all this 'hide it in your handbag' crap?
If you take a moment to read these points below from the link it becomes clear there should be proper facilities for people to dispose of insanitary waste! And not just 'wimin with period pads'. No one should be slipping sanitary waste into their handbags or the office bin FGS!
Just calmly let your manager know there's no sanitary bin in the loos and that there should be.
From the link about legal obligations:
Your Legal Obligations - What do you need to do?
If your company has any female employees, or if women from other companies visit your premises, in many cases you can break employment law, environmental law or health and safety law without even knowing it.
You may be breaking the law if you don’t provide a sanitary waste disposal bin. The ‘Duty of Care’ Act demands that sanitary waste is managed to the point of disposal. Waste must be carried by a licensed carrier and a full audit trail of documentation must be available at all times. Failure to comply can lead to a large fine or a term of imprisonment.
This applies to all commercial premises, even if only one female is employed.
This regulation is further endorsed by the Water Industries Act 1991, which states that no items should be flushed that could cause a blockage within the sewer or drain.