"how do I ensure that the colormap shows well in black and white?"
I am not sure that you can have a reliable transferrable method. You can fix this problem for these specific graphs.
Focus on the story the data is showing (here it looks like there are lines that drift lower towards the bottom right). Emphasize that location, de-emphasize the rest of the panel.
Understand you have a fundamentally 3D (vertical, horizontal, color) plot with several line traces.
Here are options that i can see.
#1. recreate the color map from a visible light/medium grey to black. the lines on the plot that appear important to me are on the 'upper' end of the color spectrum, so that should be the blackest end of the spectrum. that will ensure emphasis on the 'deviant' track lines (both shown lower on the vertical axis). The overlapping lines will be assisted by reducing opacity (i.e. 'alpha'). You will need to tune both the color spectrum and the opacity to fit the needs. All color mappings should be as consistent as possible between different graphs. This option would also require replacing the broken 'global' line with 'x' or 'o' or some other distinguishing marker shape.
#1 continued. This grey/black option might further be assisted by accentuating the log nature of the color spectrum. Currently, it looks like log base 10, you could push that log base in either direction and see how that accentuates the point in the plots you want to emphasize. You will need a bit of expertise to do that effectively. I personally would not worry about the numerical nature of the plot as much as i would the narrative surrounding it, but its not my paper.
#2 abandon color as the representation of the 3rd dimension. Replace color with height using a 3D surface plot with line tracks. This might or might not work well, depending on your comfort level, as it is fundamentally a visual illusion of representing a 3D plot on a 2D piece of virtual paper. This is a viable option here specifically because the highest 'peak' will not meaningfully obscure lower peaks when oriented correctly, meaning the graph could remain static (i.e. simple printed static image).
#2 continued. keep color and use the 3D surface height in combination. The two representations indicate the same data, but its redundant.
#3 leave the color plot as is and add an interactive element... either on this visualization or an additional visualization. For example, Plotly creates .html with hover capabilities. This way, the user can mouse hover over individual points and get a full text breakdown of whatever string you want.
#4 just say no and leave it as is. being perfectly honest, these are not the most beautiful graphs, but its not an art contest, the data is effectively represented in its current form, assuming the emphasis is on the lower right deviant tracks. if the emphasis is elsewhere, you need more than a color reimagining.
either/both #1 and #2 are likely your best bets, but both would require some amount of skill, design sense, and expertise to render effectively.
Happy plotting! gl