SELECT TOP 1 harms query performance; is there a dba-accessible way to overcome this?
If you can't change the query you can use a plan guide.

Test the performance of the query with OPTION (QUERYTRACEON 4138) (will need someone with sysadmin permissions to try this).

If that produces satisfactory performance you can apply this with a plan guide. If it doesn't produce satisfactory performance try and find a hint that does. Possibly OPTION (HASH JOIN, MERGE JOIN) if inappropriate nested loops is the problem. You might need to resort to the USE PLAN N'...' hint.

Once you know the hint(s) required you can apply them using the information here.