You can find details on the transformation from Cassini-Soldner to ARC1960 UTM in this article:
In Table 4.8, a set of four reference points is given, with an equation to transform from Cassini to UTM:
E= b * n + a * e + ∆E
N= a * n - b * e + ∆N
Where a= Scos (θ), b= S sin(θ)
N and E are the local UTM coordinates
(n,e) are local Cassini coordinates
Computed values are given in Table 5.8:
Parameter Value Accuracy Units
S 1.0000169 ±0.000002 -
θ -0.000886 ±0.000002 rad
∆N 10000167.51 ±0.35 m
∆E 277419.49 ±0.35 m
I get a standard deviation of the four points of 10 meters with those values.
Alternatively, you may use:
E = d * n + f * e + ∆E
N = a * n - b * e + ∆N
Quality of the reference points is now within 2 meters.
These transformations are valid for the Cassini Soldner projection centered on the 37° meridian, against UTM 37S based on ARC1960 datum (not WGS84!).
For the 39° meridian, transformation parameters are given in table 5.3:
Again, only valid for transformation to UTM zone 37S. Appendix 11 and 12 of the document have coordinates for both systems, and here the standard deviation is under 1 meter.