You can achieve this with some CSS
and JS
when you push the button you add the class and then you remove
Here an example to start
var map = L.map('map').setView([51.505, -0.09], 13);
L.DomUtil.addClass(map._container,'cursor');
L.tileLayer('https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png', {
attribution: '© <a href="https://www.openstreetmap.org/copyright">OpenStreetMap</a> contributors'
}).addTo(map);
L.marker([51.5, -0.09]).addTo(map)
.bindPopup('A pretty CSS3 popup.<br> Easily customizable.')
.openPopup();
#map { width: 100%; height: 360px; }
.leaflet-container.cursor {
cursor: url("https://i.imgur.com/NyfJiPA.png"), auto;
}
<link rel="stylesheet" href="https://unpkg.com/leaflet@1.8.0/dist/leaflet.css"
integrity="sha512-hoalWLoI8r4UszCkZ5kL8vayOGVae1oxXe/2A4AO6J9+580uKHDO3JdHb7NzwwzK5xr/Fs0W40kiNHxM9vyTtQ=="
crossorigin=""/>
<script src="https://unpkg.com/leaflet@1.8.0/dist/leaflet.js"
integrity="sha512-BB3hKbKWOc9Ez/TAwyWxNXeoV9c1v6FIeYiBieIWkpLjauysF18NzgR1MBNBXf8/KABdlkX68nAhlwcDFLGPCQ=="
crossorigin=""></script>
<div id="map"></div>