I am using Publisher v1.7.2 on macOS 10.12.6.
I created a very simple Publisher document--just one text box with a few words in it. Part of the text is set to Alte Haas Grotesk Regular and the rest is set to Alte Haas Grotesk Bold.
The Alte Haas Grotesk font is freely available from https://www.dafont.com/alte-haas-grotesk.font and I placed the two ttfs in a folder somewhere under ~/Library/Fonts. When I right-click on the font in FontBook and then click "Validate Font", it reports that "No problems were found validating these fonts."
Now, everything looks fine in Publisher; the font looks right, and I can see the difference between Regular and Bold. However, when I export to PDF, everything is not fine. Instead of using the Bold font, it uses the Regular weight, so there is no Bold font at all in the PDF--all the text is in the Regular weight.
I have attached the .afpub, the exported PDF, a screenshot of the PDF export settings, and a .zip containing the Alte Haas Grotesk fonts downloaded from dafont.
I created a similar document in InDesign, and the exported PDF uses both the Regular and Bold fonts as expected.
alte_haas_grotesk.zip
Untitled.pdf
Untitled.afpub