I had a look at your html code to setup a user script so I can have the requested  feature when I open the website. It's not clean, but it's acceptable for me and I haven't tested on a smartphone or a tablet, because I use only a laptop. But still, I thought I could share the code with you to give you some ideas.

- You can get a fixed podcast player by adding a fixed position to the iframe like this (look at the style part of the code)
<iframe src="https://open.spotify.com/embed-podcast/episode/0lfwKdD04ujs3MfpYbbupX" width="100%" height="232" frameborder="0" allowtransparency="true" allow="encrypted-media" style="position: fixed; bottom: 10px; right: 10px; z-index: 10000; width: 500px; max-height: 2506px; max-width: 100%;"></iframe>

- Because the player if now on the right part, you need to move the position of the text to the left by removing  "margin-left: auto;" on the .post-thumbnail, .entry-content .wp-audio-shortcode, .entry-content  (cf line 769 of style.css)

Once again, it's not "ready for production" but maybe it could help you.