Source 4 Mini LED and the City Theatrical D2 SHoW DMX Neo Dimmer

It sounds like you're picturing the S4 Mini LED as though it were a regular S4 Mini with an LED MR16 instead of the regular incandescent lamp.  If that were the case, then what you're describing might be an option.  However, the LED version uses a 120VAC driver that powers the LED directly.  Obviously the LED is driven by a lower DC voltage, but the 120VAC electronics are what controls the amount of voltage and current allowed to pass through the LED.  There's no intermediate step where a 12VDC supply could be inserted.

The inverter idea is unlikely to help you either.  I'm not aware of any inverters that would be happy running with a dimmed input supply, and the output of an inexpensive inverter is not something you could successfully use to power a line-voltage dimmer.  It wouldn't hurt to try if you've got the parts laying around, but I would expect poor dimming performance.

For your purposes it may work better to try a conventional S4 Mini with a 3rd party MR16 LED.  The idea would be to remove the electronic transformer from the fixture and power the LED directly with your battery and City Theatrical dimmer.  You may have to experiment a bit to find an LED replacement lamp that performs well with this arrangement.  You can also expect to get less output from the fixture, since the optics of LED products are different from an incandescent lamp.

It sounds like you're picturing the S4 Mini LED as though it were a regular S4 Mini with an LED MR16 instead of the regular incandescent lamp.  If that were the case, then what you're describing might be an option.  However, the LED version uses a 120VAC driver that powers the LED directly.  Obviously the LED is driven by a lower DC voltage, but the 120VAC electronics are what controls the amount of voltage and current allowed to pass through the LED.  There's no intermediate step where a 12VDC supply could be inserted.

The inverter idea is unlikely to help you either.  I'm not aware of any inverters that would be happy running with a dimmed input supply, and the output of an inexpensive inverter is not something you could successfully use to power a line-voltage dimmer.  It wouldn't hurt to try if you've got the parts laying around, but I would expect poor dimming performance.

For your purposes it may work better to try a conventional S4 Mini with a 3rd party MR16 LED.  The idea would be to remove the electronic transformer from the fixture and power the LED directly with your battery and City Theatrical dimmer.  You may have to experiment a bit to find an LED replacement lamp that performs well with this arrangement.  You can also expect to get less output from the fixture, since the optics of LED products are different from an incandescent lamp.

Additonally the LED in the fixture is not  a 12vdc component, so you would still need to do some extensive electrical gymnastics to get the LED fixture to work as your scenario would propose. The incandescent would be a better option for you.