From experience the canaries are right on the max range of the 738 with full payload from BRS on most low pressure days. If you start getting down into pressures below 990 then you start to get payload limited especially if coupled with a nice strong headwind. Tech stops in Portugal are rare but not unheard of.
Even on average days there is not a huge margin and careful number crunching is often required.
Even if you could fly a straight line to NYC, which you can't, it is still nearly twice as far to the states as the canaries. Using my total back of a fag packet 738 performance. On an average Bristol day a 738 could probably lift about 30 passengers to New York and that's working on a 50ish kt headwind but it is likely to be more on most days.
You need something properly built for the job like the 75, 76 or Dreamliner. Or a longer runway ! Can't comment on Airbus as try to avoid touching them.