This is what cox1 sequences (genetic barcodes) tell. (Data mainly from BOLD, with a few more from GenBank. No other gene has been sequenced for a large number of taxa. In particular, it may be worth to note that, although Senfeld et al 2020 used a longer alignment in their analysis, with sequences of cox1, cytb and nd2, the only American taxon for which they had more than cox1 was ater : thus the relationships among American taxa in their tree are actually based on cox1 only.)
In this tree, palliatus from Chile and Peru end up forming the sister group of bachmani; the support for this relationship may be a bit less than one might hope, and of course it's only one gene, but it seems in any case clear that these birds form a group distinct from the rest of the species. Note that there is a bird from (Pacific) Panama in the 'Atlantic' group : the separation may not be strictly Atlantic vs. Pacific.
Small distances between closely-related species are the rule rather than the exception in this entire clade. The distance between ater and bachmani may be small for two distinct species in comparison to what is found in many other bird groups, yet it is still larger than the distance between any non-fuliginosus Old World Haematopus spp. It is also larger than the distance between American and Andean Avocet, Pied and Red-necked Avocet, and Pied and Black Stilt.