You're pushing me outside of my comfort zone, you know that? I actually installed BizHawk just to try to answer your questions. Aren't you ashamed of yourself?
...Yeah, just kidding
. You did make me install it properly though, since my previous attempt had BizHawk crash the moment I loaded a ROM. Anyway, going back to your question...
BizHawk, as fellow poster Feos often said, is still being actively developed, which is why many options that might seem like common sense are missing. Saving the portion of the map displayed in the BG Viewer would be one such option. However, since the BG Viewer displays the map with a 1:1 resolution, you could sort of solve the problem temporarily by saving the content of the BG Viewer to the clipboard by selecting its window and pressing Alt + PrintScreen, then pasting that in your favourite paint program
(see 1st attached picture). You'd have to erase the rest of the windows to retain only the actual map, but it's better than nothing and with some emulators, you have to make do with what you have.
Now, just in case you're not familiar with map browsers in emulators, you have to know that these aren't true map browser as such. They simply display the part of the map that's loaded in video memory at any given time. That usually includes the portion displayed on screen, plus a bit of of-screen areas and sometimes the status bar, and you have to be careful because the map in the map browser will sometimes loop horizontally and/or vertically
(see 2nd attached picture). In short, as you, say, move your character, the game loads more of the stage map into memory. Once it reaches the edge of the "buffer" (I might not be using the right terms), it will keep loading the map by going back to the beginning (left/top) and add the new areas there. You often have to move things around a bit to recreate the real map. In this case, the clouds below the ground actually belong on top of it so if you move it there, you get the actual map that's loaded in memory at that point
(see 3rd attached picture).