Yes, you can! All you need is 2 residence proofs from these
https://www2.gnb.ca/content/dam/gnb/Departments/ps-sp/pdf/Services/List2-Acceptable-Proof-of-New-Brunswick-Residency.pdf/
And an ID proof from
https://www2.gnb.ca/content/dam/gnb/Departments/ps-sp/pdf/Services/List1-Acceptable-Identification-Documents.pdf/
In your case, the lease letter and bank statement. You do not need to wait for bank/credit card statements, can show the checkbook with the address or even better, just visit the branch and ask them to provide a letter stating you are banking with them.
By now, you would have cleared the driving license hopefully!!!
Express entry with the help of NBPNP.
Timeline as follows,
Session attended for New Brunswick PNP – May 12th, 2016
Received ITA from New Brunswick – July 7th, 2016
Sent application to NBPNP – August 15th, 2016
Received Nomination – December 9th, 2016
Sent application to CIC – December 21st, 2016
Passed medical – March 7th, 2017
Received PPR – June 22nd, 2017
Moved to Moncton (landing) – August 18th, 2017