Only saying "bar" would be ambiguous.
In Italy, a "bar" is more like a coffee shop–think of Starbucks–than a British "pub" whereas, in the US, bars normally serve alcohol.
- The 10 Most Popular Drinks to Order at a Bar (Esquire)
- 16 Classic Drinks to Order at a Bar (Town&Country)
- The Hanky Panky is a real classic, invented by the legendary Ada Coleman at the Savoy's American Bar. (Drinks International)
You do not go to a barista to get coffee, a barista is someone who makes and serves coffee and any other drink available on the menu. Despite being relatively recent, this Italian loanword is used all the time.
A "barista bar" would, therefore, suggest a bar which specializes in serving hot and non-alcoholic beverages often accompanied with light meals or snacks.
Although perfectly comprehensible, the expression “barista bar” is essentially inappropriate. The word the OP is looking for is to be found among the following: “cafeteria”, “coffee shop”, “coffee bar” or in the UK, a cafe (also café).