Assuming power levels on the broadcast are fully supplied by PoE, the range likely won't improve with the AC adapter. In the rare case, it sometimes does, but I find that to be less and less the case these days.
Despite the AC-1900P and the WAP371 using almost identical set of wifi chips (Broadcom BCM4360 and BCM43460), the radio and amplification architecture, plus power level and firmware tuning on the 1900P, probably give it the better outcome in single-radio setups (ie. most of the scenarios it's marketed and designed for). This is somewhat to be expected, whereas the WAP371 is meant to be multiplied as part of a cluster to solve range issues; ie. more, lower-power radios, versus fewer higher-power ones. Long story short, you'll likely need to purchase a second, and possibly a third, WAP371.
Alternatively, you could flip the 371 and move on to a single WAP561/71/81 and see if it has any better with range. Otherwise, at a similar price point, switch to UniFi.