Charcoal, 17 bytes

ＷＳ≔⁺ωιωＷＳ⟦Ｉ⌊Ｅι№ωκ

Try it online! Link is to verbose version of code. Takes input as two newline-separated lists of newline-terminated strings. Explanation:

ＷＳ≔⁺ωιω

Input and flatten the grid.

ＷＳ

For each word in the list...

⟦Ｉ⌊Ｅι№ωκ

... output the minimum count of all of its letters in the flattened string.

Save 6 bytes by using a more awkward input format:

ＩＥη⌊Ｅι№⪫θωλ

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

η         Word listＥ          Map over wordsι      Current wordＥ       Map over letters№     Count ofλ Current letter inθ   Grid⪫    Joined withω  Empty string⌊        Take the minimuimＩ           Cast to string            Implicitly print