Did you advise the person who told you to do it that it can breach antidiscrimination laws by discriminating against anyone with a disability that means they can’t use a mouse and that to not discriminate that you’ll have to hide the mouse cursor as well.
Presumably they will be able to afford all the court costs if someone takes them to court for breaching anti-discrimination laws.
To protect yourself you should get it in writing from them that the navigation for disabled people was deliberately blocked at their request and that you are not responsible for making their site inaccessible for the disabled. That way you protect yourself against them blaming you in court (just in case a disabled person does take such action).
but how does someone know which one will be clicked when they hit the enter key if there is no indication of which one has the focus. You need the one with the focus to change colour before that so they can tell which one the enter key will interact with.