"The restaurant is full."
Translation:Restaurangen är full.
They are not interchangeable and you can't use den restaurang here.
Den restaurangen in a sentence like this would mean that restaurant, so in that case it would be a demonstrative pronoun as you say.
This is how it works with an adjective or not:
restaurangen = the restaurant
den restaurangen = that restaurant
den nya restaurangen = the new restaurant
Other than that, there's a construction where you can use den restaurang before a relative pronoun and a restrictive relative clause, and in that case it will mean the. But it's a bit formal and isn't used that much. So you can say either Restaurangen där jag arbetar … or Den restaurang där jag arbetar… and both mean 'The restaurant where I work'. Well, or at least you can write the latter version, it's rarely used in speech.
It's because it comes after the verb. If the adjective had been before the noun, it would have been fulla: 'Vi är på den fulla restaurangen' (not a great sentence but whatever, "we are at the full restaurant"). But when the adjective comes after the verb, as in X is Y, the adjective only changes with gender/number, so that it is full for en words, fullt for ett words, and fulla for plural.