What is the quickest way or cheapest way to get into Mahogany Hall Permanently? I have managed to grind up some of the items required, but I still need 10 of each: Ridiculous Hat, Reprehensible Lizard, and Talkative Rattus Faber. I usually don't like buying from the Bazaar shop, but will if I have to. Is there another better way to get into Mahogany Hall? If not, what are the quickest way to grind the three items? Thanks.
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.
You can get those bats from stealing walls of paintings for the Topsy King in the Flit! It's 8 bats on success, if I recall. Under "Agree to Steal Paintings for the Topsy King", do the bit that covers the wall. I've not replicated the results, but my first try allowed me 8 bats.