If you're still looking, I've just gone through an almost identical
search and shortlisted the following;
KK picasso
Montecarlo
Hotel Jazz
Europark
Catelonia Catedral
Room Mate Pau
Hotel Curious
Costanza
Praktik
Pulitzer
H10 Universitat
Regina
Lleo
All ranked in the top 25% on hotels in Barce according to
tripadvisor.com, and sorted with the highest reviewed at the top. All
are sub-$250 per night for three people + infant in peak season, so
should be cheaper in October and less again if you're just a couple. If
your budget can extend a little higher, you could also consider Casa
Camper, Alma, Do placa, 1898 and Bagues. Some are closer than others to
La Rambla, but like you, I shortlisted them based on location and wanted
rooms between la rambla and placa de catalunya. Best bet to evaluate is
to search for hotels in Barcelona on
http://www.tripadvisor.com.au/Hotels-g187497-Barcelona_Catalonia-Hotels.html
using the map at the top right to see overview of the area and then just
type in the hotel names above to check out reviews and location.
Given the size of the room I was after, half of those above were not
available, but I've made refundable deposits at Casa Camper, Regina and
Curious whilst I evaluate options for the return leg of our cruise at
the end of June. For your needs, montecarlo Barcelona may be best - it's
right on La Rambla, $200 a night in peak and ranked the 8th best hotel
in all of Barcelona.
Let us know what you've gone with. Good luck!