I am looking for a way to display a fullcalenar month grid divided by Hebrew calendar months (Tishrei, Cheshvan etc.).
I managed to display the hebrew date within the cells of the month grid view using dayCellContent (including bootstrap classes):
//Array of letters representing the day of the month
const hebDateLetters = ['א', 'ב', 'ג', 'ד', 'ה', 'ו', 'ז', 'ח', 'ט', 'י', 'יא', 'יב', 'יג', 'יד', 'טו', 'טז', 'יז', 'יח', 'יט', 'כ', 'כא', 'כב', 'כג', 'כד', 'כה', 'כו', 'כז', 'כח', 'כט', 'ל',];
...
dayCellContent: (info)=>{
var hebDate = new Intl.DateTimeFormat('he-IL-u-ca-hebrew', {day: 'numeric'}).format(info.date);
var hebMonth = new Intl.DateTimeFormat('he-IL-u-ca-hebrew', {month: 'long'}).format(info.date);
//Container element
const container = document.createElement("div");
container.setAttribute("class", "container p-1");
//First row element
const elRow1 = document.createElement("div");
elRow1.setAttribute("class", 'row m-0 w-100');
const elCol1 = document.createElement("div");
elCol1.setAttribute("class", 'col p-0');
elCol1.innerText = hebDate !=1 ? hebDateLetters[hebDate-1] : hebDateLetters[hebDate-1] + " " + hebMonth; // Condition to display Hebrew month name on first day of the month
elRow1.appendChild(elCol1);
//Second row element
const elRow2 = document.createElement("div");
elRow2.setAttribute("class", 'row m-0 w-100');
const elCol2 = document.createElement("div");
elCol2.setAttribute("class", 'col p-0 c-date');
elCol2.innerText = info.date.getDate();
elRow2.appendChild(elCol2);
container.appendChild(elRow1);
container.appendChild(elRow2);
return {html: container.outerHTML};
}
This is what I achived

My next step would be to display the months according to the Hebrew calendar. I've been trying to build a custom view as discribed here: Custom view with JS, but I'm not fully sure how to use this to achive what I'm looking for.
The best option is to use the locale as hebrew. Full calendar already has that option.
Add reference to that
Combining with your code gives the result