Charcoal, 39 bytes

Charcoal, 39 30 bytes

Ｉ⌊ＥΦＥＸ²⊕ＬθΦ⍘ι12μ⁼ΣιＬθΣＥι×Ｉλ§θμ

Try it online! Link is to verbose version of code. Explanation:

      ²                         Literal 2
     Ｘ                          Raised to power
         θ                      Input array
        Ｌ                       Length
       ⊕                        Incremented
    Ｅ                           Map over implicit range
            ι                   Current index
           ⍘                    Convert to string
             12                 Using digits `12`
          Φ    μ                Remove leading digit
    Φ                            Filter strings where
                   ι             Current string
                  Σ              Digital sum
                 ⁼               Equal to
                     θ           Input array
                    Ｌ            Length
  Ｅ                             Map over strings
                       ι        Current string
                      Ｅ         Map over characters
                          λ     Current character
                         Ｉ      Cast to integer
                        ×       Multiplied by
                            θ   Input array
                           §    Indexed by
                             μ  Current index
                     Σ          Take the sum
 ⌊                              Take the minimum
Ｉ                               Cast to string
                                Implicitly print

Previous less brute-forcey 39-byte version: