Views rest export does not update
These things would be easy to solve if you could set a cache expiration time for anonymous requests in UI, but you can't unfortunately. See Page cache not rebuilding after page cache maximum age is reached
In this case you could add an additional query parameter containing the current date when website 2 is requesting content from website 1 with javascript. This would create a new cache entry on every date change in the page cache of website 1, even if the query parameter is never used. You need to disable caching of the View itself so that it doesn't get cached in other places.