Perhaps I'm misunderstanding what it is you're trying to do but there's no problem with saving a grey-scale (ie, B&W) TIFF file from Lightroom and no export to Photoshop is necessary in order to do it. I assume you're starting with a colour RAW or DNG image. Convert it to B&W in the develop module. Switch back to the library module when done. Click the Export button, lower left corner of screen. In the export dialog box select your destination folder, scroll down to the File Settings section and switch from the default jpeg to tiff using the Image Type pull-down menu. Make any other settings you need, hit the Export button in the lower-right corner of the dialog box and you're done.
The problem is that when i have finished all the editing of the photo in LR ,I want to export a TIF file in grayscale , Gray gamma 2.2 ,
in order not to have a huge file.By the way ,I think that this may interest many people who may want to send "lighter" than RGB files
(by mail for example) since the quality is not affected from the conversion.I still do not understand why this possibility is not included in LR.