BA isn't a good choice for this because it doesn't have fine and coarse thread ranges. 5BA is close enough to 1/8" and 43tpi.
Threading BSW on one end and BSF on the other would work except BSW doesn't go down to 1/8", and if it did I fear the thread would be too coarse. BSF in 1/8" diameter is exotic.
US Threads 5-40 (UNC) and 5-44 (UNF) might do: #5 is 0.125" with 40 coarse and 44 fine threads
Same idea with Metric M3 coarse (pitch 0.5) and M3 fine (p=0.35), but 3mm rod is thinner than 1/8" (3.2mm) if that matters. If fatter than 1/8" is acceptable, M3.5 coarse is p=0.6 and M3.5 fine is p=0.35.
Presumably the idea of using two different threads is to exploit the difference to get an extra fine adjustment? My tired brain remembers the method exists, but not what the rules are. For example would 1 tpi difference (5BA and #5-44) be more or less suitable than the 4 tpi difference between #5-40 and #5-44?
Dave