Google does use two versions of Googlebot. One that just fetches the page and looks at the source code. The second that renders the page.
Google will automatically use both versions of the crawler on your site. There is no way for you to flag your site in Google Search Console as requiring rendering.
The normal Googlebot crawler without rendering requires far fewer resources for Google to run. They run that crawler first and far more often. The rendering version runs later and less frequently.
That means that any page that requires rendering to index properly is always going to be delayed. Indexing and updates will both be slower. At one point Google said that the render queue was 2 months behind the normal crawl queue.
Update November 2019: Google says that their render queue is now much faster. On average the render happens 5 seconds after the crawl. In the 90th percentile case it happens minutes after the crawl. That should mean that AJAX powered sites are no longer at a disadvantage from the render queue delay.