Roxane,
I'll try in my best ability to give you a satisfactory answer, which hopefully gives you the insights that you require.
1. Birkins are never offered to walk-ins. This is THE same business model used by many other H stores around the world. You have probably read stories from ladies who scored Birkins right on the shelves, but I can assure you, these are few and far in between. Specifically in SG, NEVER. Well, at least not in the popular sizes of 30cm, 35cm and 40cm.
2. The H management is always concerned about resellers in this part of the world. And it's frown upon. So Birkins will never be offered to strangers, without any relationship with the store. And if a customer is found to have resold a bag that was from the store, for monetary gains, this customer will be blacklisted. I am not joking. This is the way it is here.
3. Maintaining a relationship at the store is beyond a Hi and a Bye. It means you have to show your patronage. There are many of us here who just love everything Hermes, and we shop at the stores. We do not hound our SAs for Birkins. When the time is right, our respective SAs will ask very subtly "Can we help you with something you want?". It's not like "Now you are entitled to a Birkin". Not like that.
4. SG stores do not just cater to just the SG local market but also huge overseas clientele especially from Indonesia. And many of them are VVIPs. No prices for guessing where most of the Birkins go to! They create a serious Birkin drought every season.
5. There is still a backlog of Birkin orders not fulfilled. And the stores no longe accept new Birkin orders.
You have specifically mentioned:-
I am primarily interested in Birkins (in slouchy leather) and also the Trim I prefer more casual-looking bags, so I am not interested in the Kelly. Nor do I like scarves or any wallets other than bill-folds and those without any adornment. Therefore, I am finding it very difficult to build up any type of relationship with a SA, simply because the rest of the items do not appeal to me.
You have actually foreclosed your outcome with the H stores. Hermes reward regular customers that embrace the whole brand, its lifestyle, its ethos. If it's just Birkins that you're after, sorry to break the news to you, you'll be better off continuing to buy from resellers.
And to answer your last question if it would help if you bring your Birkins to the store ..... NOPE.