Most of the funeral industry has been bought out by private equity interestingly. A single firm, Providence Equity, now controls most of the "Death Care" industry in the US. Also yes that is the official term.
You can transfer a domain with no downtime. The name servers will remain during the transfer, so as long as your DNS isn't dependent on the domain registrar, you'll be fine.
If you are using the registrar's built-in DNS hosting, move away from this first, which can also be done with no downtime.
As a side note, I was very impressed/delighted by the message on the site when JavaScript is disabled:
> Sorry pal, but this won't work without a JavaScript. You are probably doing that for privacy reasons, and I do respect that. I tried to handle the noscript case gracefully, but there is only so much I could do. You can download this website, inspect the source code, and run it locally. Or, if you think this website is trustworthy enough, you can whitelist it in your browser or a script blocker. I don't have any third-party trackers on this website (only a CDN), so you will be safe here.