Positioning a fake mirror for best effect
Charcoal, 91 bytes

≔⟦⟧θＷＳ⊞θＩ⪪ι ≔Ｌ§θ⁰η≔⊖⌊⟦Ｌθη⟧ζＦ⁻ＬθζＦ⁻ηζＦ²⊞υ⟦Σ⭆⊕ζ⭆⊕ζ⁼§§θ⁺ι⎇λμ⁻ζμ⁺κξ§§θ⁺ι⎇λξ⁻ζξ⁺κμ⎇λι⁺ιζκλ⟧Ｉ✂⌈υ¹


Try it online! Link is to verbose version of code. 0-indexed. Explanation:

≔⟦⟧θＷＳ⊞θＩ⪪ι 


Input the mosaic. (These 12 bytes could be avoided by requiring the input to be in JSON format, but I was too lazy to punctuate the example.)

≔Ｌ§θ⁰η


Get the width of the mosaic.

≔⊖⌊⟦Ｌθη⟧ζ


Get the inner size of the mirror, i.e. the distance from the first to the last character of the mirror in terms of diagonal steps.

Ｆ⁻Ｌθζ


Loop over the possible row(s) of the top left corner of the mirror's enclosing square.

Ｆ⁻ηζ


Loop over the possible column(s) of the top left corner of the mirror's square.

Ｆ²


Loop over the possible rotations of the mirror.

⊞υ⟦Σ⭆⊕ζ⭆⊕ζ⁼§§θ⁺ι⎇λμ⁻ζμ⁺κξ§§θ⁺ι⎇λξ⁻ζξ⁺κμ⎇λι⁺ιζκλ⟧


Calculate the number of tiles it reflects exactly. Exact matches are counted twice and the diagonal is also counted but this doesn't affect the relative score. Push this number along with the potential solution to the predefined empty list.

Ｉ✂⌈υ¹


Output the solution with the highest number of exactly reflected tiles.