Self-publishing print on demand (POD) services like KDP Print and IngramSpark offer two papers: White and a slightly thicker "creme" paper. It's quite nice - I used it for a fiction series - and the cost is not much more than white. But it's not the same grade as the Bible paper you mentioned.

If you work with an offset printing service you can do a short run on whatever kind of paper you like but the cost will be more for smaller orders of 250. At 500 you start to see economies of scale that rival POD costs but you have to take shipment of hundreds of books and store them somewhere.

What your looking for is called onion skin paper and it seems to be pricey, but I agree, great for certain uses

I’m willing to pay. Do you know how to get a book bound using that paper?

Like ilamont said you’ll probably have to go through an offset printing process. There are some online services that offer offset printing for flyers etc. but only with the most common setups. Easiest is to find a few printing companies in your area and ask for offers. You might ask book designers or publishers in your area to recommend printers.