Substituting almond milk for regular milk in coffee without bitterness
I don't have a confident answer as to why it's happening. My wife uses Almond Milk and drinks it with her coffee. She's never noticed any bitterness. My first suggestion would be the coffee itself, rather than the almond milk. The way you make coffee (in a moka pot) will produce different results than at Starbucks or any other coffee shop themselves. That said, if I understand you correctly the local coffee shop you're drinking at isn't Starbucks. Personally I find many of their beans to be bitter... Despite you not being able being able to identify the bitterness in the coffee when drank black, I'd suggest that those beans are your problem. Perhaps the Almond milk is highlighting out some unpleasant flavours in those beans. 

Things to try (Starting with what I'd guess as being the most likely to solve this...):


Buy beans from your local coffee shop, preferably the same ones they serve. Ideally this coffee shop can tell you when the beans were roasted, and preferably this date is recent-ish. But, if not, that's fine, Just try a different bean first.
Add a bit of salt. See this related question
Change the way you make coffee (drip, moka, etc...) for a day and see if the harsh tones are still present when you add your Almond Milk
Do you use your moka pot regularly, if you hadn't used it in a while, perhaps there are some harsh flavours that have gone stale? (Kinda stretching here).
Try a different Almond Milk?
The last thing I would do is add sugar... That usually doesn't cover up the bitterness.