Formula is changing at Save
Here is the formula as Written:
=INDEX({Req Main Sort Summary}, MATCH($[LIBID]@row, {LIBID}, 0), MATCH([Script]$1, {Req Main Sort Header Row}, 0))
Here is what happens when I save it:
=INDEX({Req Main Sort Summary}, MATCH($LIBID@row, {LIBID}, 0), MATCH(Script$1, {Req Main Sort Header Row}, 0))
This is consistent, it will not hold the formula as written.
Answers

Column names that do not have spaces, numbers, or special characters do not require square brackets, and Smartsheet will automatically remove them. It will not cause an error.

Thank you. That answers part of the question, of course. I had to go back and look at a couple of my other cross sheet Index/Match groups instead of relying on the one I was using. I am still getting a #INVALID REF result that I cannot find the answer too, which I thought was being caused by the above problem. Now I am at a complete loss.

That particular error means that you have a {Cross Sheet Reference} that hasn't been set up properly.
The easiest way to figure that out is to click anywhere within a {Reference}. In the formula helper dialog box that appears when typing out formulas, the little blue link should read "Edit Reference". If it reads "Reference Another Sheet", then you will need to remove that {String} and follow the appropriate steps for inserting a cross sheet reference.
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