Everything is possible with Custom CSS. that being said you have three Menu styles available, primary, secondary and extra. SO given you are using an extra menu shortcode for the desktop menu, you can use a different style for the extra menu than for the primary menu bar.
Either use the secondary style or the extra style.
That will allow you completely different styling for the desktop menu and the mobile one. The only thing left will be to change the Current page styling.
Once you have both menu styled with their own style, tell which one you want to change and how and we can give you the CSS