I am not sure if you can call this a bug. My definition would be that x11 handles such things different than Wayland.
To help you and without telling you “use Wayland instead”, I would like to know what keeps you using x11?
Have you tried to put the left screen (in settings) on the same height as the big one?
This could be the reason not seeing the top bar.
And you have to switch the option on to see the top bar on both screens. I guess you made that?
When you have dual screen, you mostly have the option to choose on which screen do you want the Top Bar alias menu Bar.
Som DE’s give you the option to switch it on on both screens.
p.s.
It looks like you are still using x11? Or do you have Wayland in Budgie too?
I like to have the icon bar on my laptop screen and use the main screen to do my work. I don have X11. I don´t believe there is a budgie live with wayland?
Which leads to antoher question: Apart from being newer is wayland better?
The definition of “better” depends on the use-case. Xorg has become a maintenance nightmare. Many developers with have in-depth understanding of Xorg are aging out of the workforce and AI projects are paying well for linux devs, so the pool available to work on Xorg is shrinking. Xorg lacks isolation between windows (you can demonstrate this with xeyes in Xorg versus Xwayland). This is a security problem but also something that some Xorg applications use.
If your hardware and apps support Wayland, you avoid some known security risks and can expect more attention to maintenance going forward.