Your question asks if/how "the body will adapt its production of cholesterol".
Literally, any cell - that should include the hepatocytes of the liver - of "the body" produces cholesterol, as it is an essential part of the membrane (of animal cells, by the way, by a factor of 100 more so than of plant cell walls), and it should be empirically well ascertained that there is adaption of cholesterol production to dietary input (reflected, especially, by delivery of cholesterol by - so called "remnants" of cholesterol transporting lipoproteins and chylomicrones, which may very much seen as a regulatory system that balances production between specialized cholesterol producing cells, i.e. especially hepatocytes of the liver and adipocytes of the tissue).
However, refering to any mechanism your question may be about, there are still open questions. See quote below, last sentence.
Here's the only pertaining quote I found.
Takizawa, Cellular control of cholesterol:
"When cells are faced with low cholesterol levels, they respond by increasing the gene expression of proteins that stimulate biosynthesis of cholesterol, such as HMG-CoA reductase, and proteins that increase the uptake up cholesterol from the external environment, LDL receptor. These genes contain a common upstream regulatory element called the sterol response element that binds a transcriptional activator, called the sterol response element binding protein.
When bound to the element, the binding proteins turns on transcription of downstream gene. So, a key question is how the cell regulates the binding of the binding protein to the response element and how that regulation is sensitive to cholesterol levels."