Hi JM. Lots of interesting sidelights there; not sure that any of it applies. T Minton was in NZ until 1872 and then spent a couple of years in Victoria, so would not have had much to do with Christian Hense or Dr. Bailey's marriage Mill.
Perhaps if you checked out this link:
https://sydney.edu.au/paradisec/australharmony/register-M-2.phpit would give you a clearer idea of the type of musician TM was. I don't think Street musicians or German Bands quite fits the bill. He was also a teacher of music; taught for some time in Dunedin (see ads on Papers Past) so renting from music teacher Hense probably makes sense.
I agree ETF was not charged with brothel keeping; but given the comments throughout this thread that Exeter Place was a pretty grim area, it's hard not to reach that conclusion, especially as the act under which he would have been charged is specifically aimed at controlling "infection and contagious diseases" and he could have been required to whitewash the entire area (see Hansard). Especially as the labour value of L2 today is $1798 - not such a great amount, I imagine considering today's wages.(
https://www.measuringworth.com/calculators/australiacompare/relativevalue.php.
I checked out the archives you recommended but the relevant ones are not yet digitised so couldn't come up with any additional names and since I don't live in Sydney (or NSW for that matter) it's going to be a while before I can do anything more on that front. In the interim I've checked out all of TM and MM's children. BDM records all show name of mother and father. And, of course, In memoriams went into the SMH for a couple of years after TM's death. Mother's name definitely known but no-one wanted to recognise her.
As for the preponderence of Thomases - at least two surnames, Letitia's husband and the German Band - I guess with Sydney's population at around 29000 in that era and many of them crammed into the small area of the Rocks, Darlinghurst, Surrey Hills, etc, I suppose it would be hard not to have multiples. It certainly seems like that when doing a name search on Trove.
Harking back to other valid points you've made. The trove article that seemed to mention Letitia when I increased and cleared it became Hortensia

;
and in case you haven't been able to find it, in 1889 TM apparently rented 2 places in Rockdale; it appears in trove as notification of assessment (SMH18/5/1889) Lot 1, Sec 1, and Lot 36, Sec 1, Hastings, Ellery's sub-division.
Also, The Age (Melb) 23/06/1875 TM advertised that he would not be responsible for any debts his wife might incur after that date. Their last daughter was born 28th April, 1876 at No. 2 off Washington St, which just fits the appropriate time for M to have returned and an infant to be conceived. No syphillis with that one but no wonder they left Melbourne. I suspect after the third time he really meant it.