Which profile should I choose in output export options? I need to print on office printer with clean grey colors (without CMYK tints).
You are correct that, without a postscript RIP, most end-user inkjet printers expect to see RGB data and many have no clue what CMYK color data is. This causes the printer to convert the unknown color (CMYK) to RGB... then convert the RGB to CcMmYyK for printing. So yes, you get 2 color conversion which in some scenarios can alter colors.
To export an RGB PDF from InDesign, merely choose a job option like "High Quality Print" Then adjust the
Output options... Color Conversion set to Convert to Destination, and then setting the Destination profile to an RGB color profile.
This will use Acrobat to convert from your CMYK InDesign file to an RGB PDF.
sRGB IEC61966-2.1 is a fairly standard RGB profile, but you may also have an RGB profile specific to your printer. If you do, use that.
You don't have to do anything. Office printer are not usually set to print CMYK files. Even If they have CMYK toners or paints it will change it to RGB and then to CMYK profile of the printer.
So if you want to print grey form K it should be a setting in the printing options or printer setup.