"West"? I take it that would mean south(ish) of the State St. grade crossing where those two stubby low platforms currently sit, since that's the straightest area of track close enough to the station building and there's a decent amount of slack space on the ferry terminal side to: 1) lengthen both the side and the island platforms, 2) reconfigure as 2 side platforms so SLE and Amtrak stay out of each other's way, or 3) straighten the tracks slightly. That would work pretty well.
"West" / north(ish) at the Winthrop Blvd. crossing would make comparably less sense being further from the station building, having far less width to work with on platform construction, and being on the 3-lane part of Water St. that's much wider and busier to cross than quiet back alley S. Water.
Still, I can't imagine it's 100% physically impossible to construct gapless full-highs in front of the station building. The MBTA's Greenbush Line has stations with 800 ft. full-highs on pretty tight curves
that don't have problems with platform gaps. Now, I don't know what the precise difference in degree of curvature is between that example I linked vs. NLN, but it's close enough that a little finessing to knock off a couple degrees ought to do it and keep a unified station at/across from the station building (and where the pedestrian overpass is going slated to go when the new museum parcel gets developed on the water-facing side of that block). It may require doing away with that runt of an island to go with a second side platform, track-shifting, and reconfiguring the crossovers so freights and any Amtrak expresses pass in the center. But the work involved seems to be more more about tackling the requisite design precision
(and funding therein) rather than yes/no is it feasible at all. Unless there's just something different about the dimensions or door configuration of the Mafersa coaches, M8's, and (possibly) Amfleets that exacerbates the platform gap situation in a way the MBTA has no problems with at its sharp-curve full-highs with its bi-level and Shoreliner-clone single-level fleets.