Actually you both are looking at SJM's diagram, not OP, and SJM's is just fine, and he's probably only about 3' from wall.
The joists should be plenty to carry load, beam is more of a brace to keep from sagging over time, and just added assurance.
I'm a GC, I remove or open load bearing walls often.
Common code here is typically 4"x6" for a 6' opening, 4"x8" on an 8' and so on.
Since this is just a brace added to joists, you could probably go w/ something like a 4"x8" or 4"x10" sitting on top of 4"x4" posts at each end against the walls as long as you have access to squeek in beam and span isn't super huge.
Not even a full day to do including PU of material.