Case-insensitive is indeed the correct antonym. That's the normal tech term. Evidently, it can be used with or without the hyphen.
To cause a case-sensitive comparison of nonbinary strings to be case insensitive, use COLLATE to name a case-insensitive collation.
This example shows that VERSION() returns a string that has a case-insensitive collation, so comparisons are case insensitive:
As for the adverb, to me case-insensitively sound a bit awkward, so I would prefer in a case-insensitive manner:
You can configure the property value comparisons in query search conditions to be performed in a case-insensitive manner.
Or, you can simply do without an adverb or adverbial phrase and write something like:
This will perform a case-insensitive comparison.