Fallen London: Quickest Way to get into Mahogany Hall?

The only other way to permanently unlock Mahogany Hall is to pay 5 Fate on the opportunity card "The Ways of Mahogany Hall".

The TL;DR is you're just going to have to buy those items from the Bazaar.

It's possible to get Reprehensible Lizards from rare successes on the Investigating... carousel in the Flit, but the chance of a rare success makes it very much not worthwhile unless you're really, REALLY not in a hurry.

Similarly, Ridiculous Hats can be obtained from a rare success while "Gambling for silver" if you're on the Heart's Desire! Ambition and don't have the 77 First City Coins required for it. Again, the chance is extremely low, so the time required to get 10 makes it inadvisable.

Talkative Rattus Faber can't be repeatably acquired anywhere other than buying from Nassos Zoologicals at the Bazaar.