This is an extremely hacky solution, but it works for me:
h1 {
page-break-inside: avoid;
}
h1::after {
content: "";
display: block;
height: 100px;
margin-bottom: -100px;
}
Basically I create an invisible element that increases the size of the <h1> without affecting the content after it. When page-break-inside: avoid is applied and the whole <h1> (including the hacky element cannot fit into the page) the browser is forced to carry the <h1> to the next page.