I normally don't reply to petitions for "which restaurant" as I find it something very personal: too often we all have very different tastes, different past experiences, different budgets... and different expectations! And also because this is a city with a very fluid gastronomic scene with several thousand eateries where you can find a few true gems and many crappy places too... and where restaurants are popping up and winding up all the time. Yet today I'm soft, LOL!
Adapting to your itinerary...
Tue > Platja Ca la Nuri, by the beach https://platja.calanuri.com/en/
Wed > Tapas24 in Carrer Diputació http://www.carlesabellan.com/mis-restaurantes/tapas-24/
Thu > Kibuka in Carrer Goya in Gràcia http://www.kibuka.com/en/goya-eng/gallery-goya/
Fri > El Xalet in Montjuïc, with splendid views over the city https://www.gruptravi.com/en/xalet-montjuic/
Sat > Tapeo in Carrer Montcada http://www.tapeoborn.cat/tapeo-del-born-en.html
All of these are no-frills no-nonsense casual places... except for Tapas,24, which is still casual but it's run by a famous trendy chef and El Xalet, which is a bit posh and it's probably the priciest of the list -still, it's "reasonably pricey".
Although not strictly necessary, you might want to call them and make reservations as August is especially busy.
There's another one particularly coming to mind, Can Recasens (https://canrecasens.restaurant/), located in El Poblenou, a neighbourhood outside your itinerary, yet still very worth visiting, which I like a lot because it encompasses a few of my favourite requirements: casual, cosy, inexpensive, romantic, not noisy, not touristy, very local and with good quality simple products. This is a good place to drop by for a light dinner. The place was originally a charcuterie back when the great-grandparents of the current owners opened it in 1906 and today they keep the tradition as they've turned it into a restaurant serving tables of local cheeses, pates and cold cuts and salads: simplicity yet good quality. This sort of eating is very typical here in Catalonia when going out with close friends. For this place, you do need to make a reservation as it's very popular among locals and it's almost always quite full. You can reserve for a table either at 9 pm or at 11 pm. You can go there by metro or in an inexpensive 15' ride by taxi from the centre.
FYI... El Born is not a neighbourhood, nor a district... it's just a short 200yd boulevard. The neighbourhood where is located is called La Ribera and the district is Ciutat Vella.
Enjoy
PS: Check the sites and follow your guts... I make no promises nor accept any responsibility :)