K (ngn/k), 14 bytes

{&/'(#'=,/x)y}

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Takes the grid of letters as x (the first arg), and the list of words as y (the second arg).

• (#'=,/x)
  • ,/x flatten the grid of letters into a single string
  • #'= build a dictionary mapping the unique letters to the number of times they occur
• (...)y index into this with the list of words
• &/' get the number of times the least common letter in each word appears in the grid (and implicitly return)