I don’t know.
I’ve had another think about this. There’s another reason for the low current switch off, and that’s because it is really hard and therefore expensive to design a 3.7V to 5V DC converter for 2.1A or higher and still have it properly regulated at less than 100mA of load. Factories that make these devices can’t afford to make one that costs too much, so they don’t.
The power bank products are designed for the mobile phone and tablet recharging market, and not the kind of things we want to do with them instead. We’re frankly lucky to have Adafruit go into detail on the low current switch off; that detail raises their risk, and that’s their choice. They will have added that into their price, no doubt. That Jaycar/TechBrands have not is emergent from their restocking practices; minimal product detail lets them buy in a different device later, lowering cost.
Thing is, stock is not guaranteed to be all-the-same outside the product description. It will be functionally identical, but not necessarily internally identical. Unless the product is described as having a minimum, the stock could vary quite a bit on what minimum power must be drawn to keep it on. I’d be worried that the Adafruit product might need you to draw more or less power to keep it on when the temperature is very different to normal.
If the product has a display, or a button, there’s a very good chance it will have a low current switch off feature. If the product contains more than one lithium-ion cell, it will almost certainly have a low current switch off feature, because voltage below minimum is very unsafe for multiple-cell parallel packs; it can lead to cell imbalance with no practical way to overcome it. If the product looks to be the size of a single 18650 cell and doesn’t have a button, the chances are good that it will do what you want.
In your application, if you propose to draw less than 100mA then I suggest a Pololu 5V DC step-up switching regulator attached to the battery terminals, with a separate USB socket. That way it won’t switch off, and won’t be glitched when the charger turns on. You can also build up such a battery subsystem using bare cells with a solar charger.
For remote dark sites, you might also worry about daytime maximum temperature. Lithium-ion cells degrade quite quickly above 40°C, your best bet are the Lithium-Ferrophosphate (LiFePO4) cells, with a lower charge voltage.
Little Bird has a returns policy; you can use it to get a good look at a product, but read the terms and conditions carefully. It has changed since I last used it.
Hope that helps!