If you can print the colored pages separately and insert them manually, simply print the colored pages by a good inkjet printer with black included and then insert those pages into the piles of BW laser outputted pages.
This is not a nice solution, if nearly every page has a little color, altough they are mainly BW.
Even in that case it should be easy to make two printable versions of your work: Version 1 with the BW only objects and version 2 with colored objects only. Then print Ver1 in laser and overprint Ver2 in color inkjet.
What makes your original idea difficult - a little background:
Many office printers unfortunately take RGB input. The RGB is converted to CMYK in the printer without letting a possiblity to input a non-K version. This system is good for office, because Word, Excel etc... have no idea what is CMYK.
As far as I have used office printers, there exist no possiblity to cancel C, M, Y or K in the settings or printer driver. I've tried it also in Adobe Acrobat, but it simply refuses any discussions about color separation, if the final output printer is RGB.
I've met a case that some ink has runned out. This can in theory be your solution. Unfortunately the printer can prohibit the printing at all when some bottles are empty or removed and if it allows to print without some bottles, the printer can get damaged. But still you must generate a K-only image for BW laser printer, becauseI do not believe that the system adminstrator lets you remove C, M and Y cartridges from a color laser printer to save powder.
The idea to get the K for CMYK from BW laser is in theory possible by making a partial color separation. In Photoshop simply have an image as CMYK and turn off C, M and Y channels - only three clicks required. This is your K-image. Your CMY image is full CMYK image with K-channel filled by white. But you must convert your pages to high resolution images. A hefty job for long document amd high need of storage until you print as you go and delete all used K and CMY images.
It, of course can be automated as actions. If your pages are exported as separate one page CMYK images (JPG, TIFF, PDF) from InDesign (a few clicks exports all pages as one page files), you can simply record the Photoshop processing for one and repeat it automatically as an action.
How good is the color? Probably some adjustment for K.images are needed, because the density of BW laser black differs from inkjet's. But that can also be automated after a proper adjustment is found.
Addendum: I have no idea, how well two different printers print into the same position. Overprinting definitely is not a major design goal for office printers, but in theory proper convergence still is possible. Check the convergence before you burn more time to the partial color separation + overprint -idea.
Another lately born to-do is to check, what laser's black (=melted plastic) affects to paper's inkjet-printability. Test run: Print some color over laser's dark grey Does the ink stick?