OK, tricky to make that element sticky and not have it up most of the screen.
How about this.
Keep what you have added so far.
We’ll add a second button that will be part of the sticky navigation on mobile?
If thats good then:
1. create a new Block Element in Appearance > Elements.
2. The Element Type should be Hook, and the hook should be menu_bar_items
3. In the block editor add your button with the call number.
4. Select that Buttons block ( note that the Core Buttons Block is a buttons wrapper and a button inside, we want the wrapper ). And in Advanced > Additional CSS Class(es) add: hide-on-desktop hide-on-tablet sticky-call-button
Once thats done it should appear in the Mobile nav, let me know and ill provide some CSS so it only appears when its sticky.