You are going to print it and that happens by using some CMYK printing process, only leaving out cyan and yellow inks, I presume.
Your best way to get it right is to use a system which can create and show CMYK files and respects the forthcoming CMYK printing process when it shows your file during you make adjustments.
Nobody here knows as numbers what's your "best match". The only way to get your "best match" is that you make some adjustments by yourself to the magenta and black color channels until it looks right.
Unfortunately GIMP is useless for plausible preview of the result. Someone may shout "False! GIMP 2 has color managed soft proof view mode for common CMYK print color profiles!"
Yes the preview mode is there, but you can adjust in GIMP only the RGB image. The preview assumes certain conversion method from GIMP's RGB to the wanted CMYK profile. You can select some principles how the image will be made dull enough for CMYK printing. They let you select the rendering intent.
But the resulted CMYK file is inaccessible, it's only calculated in the computer memory and it stays out of your reach. In addition it definitely contains something also in cyan and yellow for the best color accuracy. You have no way to limit the CMYK printing to magenta and black inks and see what they give to you. Mixing magenta and black layers in an RGB image (already suggested as a workaround) do not make it. Soft proof preview assumes cyan and yellow inks are also available and uses them.
You may wonder why in the hell you should use any printing process dependent program? Shouldn't it work if one adjusts and blends in the simple way (=multiply) colored and black RGB layers against the white background layer until the result looks right?
Yes. It works in deciding what you want, but it knows nothing of the non-linearities of printing processes. After getting your 2 print plate RGB images and a RGB image of the wanted result the printer may do his best effort to create the needed print file and sends to you a soft proof preview. If not, he either cancels the whole job or prints something which is not what you wanted.
Obviously you are not going to rent Photoshop to do the job. But you can get another freebie: Krita. It understands CMYK printing and can handle internally the images as CMYK files.
You should convert your photo to grayscale in RGB mode in a way which more or less (if wanted) changes color hue differences to brightness differences. This allows having some contrast also in cases where the straightforward desaturating would cause a foreground object to vanish in the as bright background. In old days photographers used color filters to get it. Now you can desaturate with color weighting. Krita's preinstalled G'MIC filter pack contains the perfect desaturating tool for it:
The image is still in RGB mode. I adjusted it to make the red coat bright to keep it well apart from the horse behind. This maybe is too much, but you can judge it by yourself:
In the left the brightening of the red coat is used, in the right the desaturation is uniform. Both images are CMYK tiff files shown in Photoshop (it's Photoshop only to double check they really are CMYK files which have no ink in cyan and yellow channels and soft proofing really looks the same as in Krita).
This is the desaturating in G'MIC:
It's called normalized desaturation, because it automatically keeps the overall brightness the same. For that reason there's no weight slider for green.
In Krita, after desaturation, it's the time to colorize the image. This can as well be done in GIMP, because it can be done in RGB mode. I inserted a solid maximally saturated magenta top layer with blending mode overlay. As well one could use a colorize filter. Black stays black, white stays white and 50% bright grey get the most colorful magenta:
Merge the image to a single layer if you used layer blending to colorize it.
In Krita: convert the colorized single layer image to CMYK. Its Image > Convert Image Color Space. I selected CMYK color profile Euroscale Coated v2 and perceptual rendering intent (no screenshot) That's a random selection. Your printer should tell what's good for his print process. See NOTE.
The next step is to turn ON View > Soft Proofing to see more accurately what the printing will give. It's surely duller than it was in sRGB. But the difference is not dramatic because there's not much other bright colors than magenta.
The image contains still some cyan and yellow. That's automatic for more linear looking grayscale, but those inks must be removed. Open Filter > Adjust > Color Adjustment Curves:
At first turn cyan and yellow channels to zero. Then adjust magenta and black channel curves for good white-magenta-black balance. It's not only an opinion. The printer may recommend how much there can maximally be black and magenta ink together at the same time. (see again the NOTE). In the next screenshot the magenta curve is adjusted so that there's at least a little magenta everywhere, but there's nowhere full 100% magenta.
Save the file as TIF. Remember to include the color profile:
I'm afraid the printer wants a proper print PDF. Scribus is a free tool for making one.
NOTE It's well possible that the printer doesn't tell anything of the good settings if you do not ask. The person who takes the jobs from the customers maybe doesn't even understand any details, so you must be in touch with a person who knows them.