Thanks for the feedback Bob and Echizento. I have another double-axe that I hope to be posting soon.
Further searches into Abbaitis didn't turn up much of interest, apart from the description below from the Dictionary of Greek and Roman Geography. I don't think it clarifies matters much though.
"A town of Phrygia Epictetus. Strabo (p. 567) calls it a "small city, or hill-fort, near Blaudos, towards Lydia." In another passage (p. 576) he says that the Rhyndacus, which flows into the Propontis, receives the Macestus from Ancyra Abasitis. Cramer (Asia Minor, vol. ii. p. 12) corrects Abasitis into Abbaitis, on the authority of the coins and an inscription found in these parts. As the Macestus is the Susuglierli Su, or the Simaul Su, as it is called in its upper course, Ancyra must be at or near the source of this river. The lake of Simaul is the source of the Macestus, and close to the lake is "a remarkable looking hill, the Acropolis of an ancient city." This place appears to be Ancyra. The river flows from the lake in a deep and rapid stream; and no large stream runs into the lake. Simaul seems to be a corruption of Synnaus, or Synaus, and to be on or near the site of Synnaus. Ancyra was on the lake, 7 or 8 miles WNW. of Simaul. (Hamilton, Researches, &c. vol. ii. p. 124, seq.)"
I have another coin from Abbaitis, but didn't realise it was from the same place at first, as it was recorded as Abbaitis - Mysia. Wildwinds shows four bronze coins from Abbaitis on the page titled, "Browsing Ancient Coinage of Phrygia, Abbaeti", then lists them as, "Mysia, The Abbaeti", and "Abbaitis, Phrygia", to add to the confusion. I'd like to find one with the head of Herakles in lionskin.
I've had this one for a while, so it still has the black background.

Abbaitis - Mysia, Zeus / Thunderbolt. 2nd century BC.
Obverse: Laureate head of Zeus right. Reverse: MYΣΩN-ABBAITΩN above & below thunderbolt, all in laurel wreath. Bronze. Diameter: 19 mm. Weight: 5.9 gr.
Reference: BMC 1; Mionnet 2; SNG Cop 1.