2. Initial page load will be slightly higher (and subsequent pages will be much faster) but I can't imagine it'll be too noticeable. People get away with a lot - even on mobile - loading numerous JS libraries, tonnes of CSS, etc. As long as you go easy in other regards, this shouldn't be a problem.
There's no reason that API driven design has to involve client-side rendering. You can always consume your API on the backend and render on the server.