The accession date of Shah Jahan III is 08 Rabi II 1173AH (29-11-1759) and deposition 29 Safar 1174AH (09-10-1760). Shah Jahan III ruled less than one year (moon as well as solar year). Correct dates for this ruler are therefore 1173/Ry.Ahd and 1174/Ry.Ahd.
There are coins of Surat mint in his name with curious Hijri dates.
I have requested Richie, to add the images of two coins below:
1) Rupee Shah Jahan III, mint Surat, AH117(1,2,3, 4 or 6)/Ry.Ahd. The Hijri date looks like 1171, but this can’t be correct and most probably might be a 3 or 4. However a 6 could also be possible, particular as there exist coins with later dates.
2) Rupee Shah Jahan III, mint Surat, AH1180/Ry.Ahd. This is a most curious date as by that time we right the 7th or 8th regnal year of Shah Alam II. Who was responsible for the issue of this coin? Imitating the mint name on coins is not a novelty in India, it happened very often. The pseudo mint name Shahjahanabad was used by many issuing authorities, from Jaisalmir to Maratha mints like Bagalkot. Therefore I suggest that the rupees of Shah Jahan III of Surat mint, with the later dates, were issued by another authority than the the Nawab of Surat.
A neighbouring State issuing copper coins in the name of Shah Jahan III is Bhaunagar, even up to 1825 (see C#15b, which coin is a clear imitation of a Surat issue too, rf. KM#218.2). Bhaunagar is located right opposite Surat on the Gulf of Cambay. I presume that this may be a reasonable candidate. The British struck rupees with the mint name Surat at Bombay, why not Bhaunagar had Surat rupees struck at their own mint too?. The copper falus are all in the name of Shah Jahan III, so one would expect the same for the rupees.
This is my personal presumption and I have not yet been able to prove it by contemporary records or so, but it is worth looking into that direction.