Prevent SQL from apparently executing an expensive sub query
If you add OPTION (RECOMPILE) to the very end of the WHERE clause so it looks like this, does the procedure run faster now?
WHERE A.DateValue &gt; '2020-10-01'
    And 1 = CASE @SearchString
        WHEN N'' THEN 1
        ELSE
            CASE @ComparisonType
                WHEN 0 THEN -- potentially expensive query 1 referencing TableC
                WHEN 1 THEN -- potentially expensive query 2 referencing TableC
                WHEN 6 THEN -- potentially expensive query 3 referencing TableC
                WHEN 8 THEN -- potentially expensive query 4 referencing TableC
                WHEN 10 -- potentially expensive query 5 referencing TableC
                ELSE 0
            END
        END
OPTION (RECOMPILE)

If so, then you likely have a parameter sniffing issue which could be causing bad execution plans to be generated with poor cardinality estimates. If you update your post with your execution plan, then we can pinpoint exactly how that's happening.
If that's the case, this is a good article on Parameter Sniffing and OPTION (RECOMPILE).
Note I usually only use OPTION (RECOMPILE) for debugging to quickly confirm a parameter sniffing issue and try to resolve the underlying problem instead. There are use cases for OPTION (RECOMPILE) however it's a trade-off for minor performance hits since the query gets recompiled every time it's executed.
Brent Ozar has an awesome series of blog posts on Parameter Sniffing issues. Too many to individually link to be honest, so here's the root page that links the main important ones: Parameter Sniffing