Why Aren't These LED Bulbs Lighting?
With LEDs, 12V dimming works differently from mains dimming.  

The pictures of these LEDs show they are set up just like 12 volt LED strips - 3 LEDs in series with a resistor, repeated as needed.  They also havea bridge rectifier, the angled devices in the photos, to assure they work in either polarity or on AC.  

AC Mains dimmers for incandescents typically use trailing-edge triac dimming, which manipulates the AC sinewave.  Mains AC LEDs "dim" the same way. Actually they are immune to dimming, but they have some intelligence that reverse-engineers  what the triac dimmer  is trying to do, and they dim to match. 

Your LEDs are not capable of this.  That doesn't make them unfit for dimming, but rather, unfit for mains style triac dimming. 

In the low voltage world it is quite different.  In incandescent, there are several ways to dim, including reducing the voltage, or triac dimming again.   However, LEDs use a technique called PWM, which turns the power on and off thousands of times a second (not 100 times a second like triac dimming).  

These LEDs are built to be compatible with PWM dimming. They might work somewhat with triac dimming, but I'd expect them to be shimmery since they turn instantly on/off. The seller didn't mention this because they are simply hawkers of cheap Cheese junk off Alibaba, which is direct shipped from China (or Amazon's fulfillment warehouses) to circumvent your country's quality and safety standards.  That said, those LEDs are so simple that they should work electrically even if they do fill your house with carcinogenic smoke at the first power spike. 

This luminaire's dimmer is not only a triac dimmer, but is the old style that connects in series with incandescent lights. To work, it must flow a certain amount of current through the incandescent whether it's on or off, and your LEDs + 1 incandescent aren't capable of that. You could try inserting more incandescents, but that would ... defeat the purpose ... 

The smarter play is to rework the luminaire to use PWM dimming.