Are you printing from a PDF created from your InDesign document?
As per your query when you print from InDesign colors are inconsistent. Could you please check how colors look on screen if your export file to Print PDF?
Please try these steps:
File > Export > Print PDF > Choose PDF/X-1a:2001 PDF preset
When colors shift so dramatically, it usually indicates a problem with color management (what you see on the screen versus what is printed on various printers). Also, converting from RGB to CMYK can cause this, too.
InDesign Secrets Magazine has some good basic instructions about color management that you might find helpful: https://indesignsecrets.com/indesign-color-management.php
and this recent post in the forums discusses similar issues: Color management issues in PDFs destined for desktop printing
You can choose a profile that conforms to FOGRA 39 or Fogra 51 standard.
Simply select the relevant option from the Bridge Color Settings settings and sync.
Profiles are a very comprehensive issue. It's enough to use the standard settings.
If you are using the RGB color mode, your colors will always be inconsistent. In Photoshop you should use the correct profile when converting from RGB to CMYK.
When you command Print in Indesign, you can also select which profile you want to print.
I use the / Europe Prepress 3 option via Bridge. When I synchronize, the settings are synchronized in InDesign, Photoshop, and Illustrator.
I'd like to be able to use the correct printer profile in Indesign but not sure which would be the best to use.
I'm using an HP MFP M278 printer, and I also have an Epson Stylus Pro 4900 for posters.
Documents printed on desktop printers should be in RGB, these printers do their own conversion to CMYK.
Printing CMYK documents will lead to a "double conversion", and may result in wrong colors.
When you install a printer, several printer profiles are usually installed, and you should pick the one with the correct printer model - paper combination. If you use a non-Epson paper on the Epson printer, check if the paper manufacturer has a profile for your printer. If not, you will have to experiment, and try to find one that works.
Alternatively, you can let the printer manage colors, this may also require some experimenting.
I am not familiar with the HP MFP M278, but it seems to be quite basic, and as far as I can tell, there aren't any printer profiles for it.