On-the-fly query/command modification
This is not a technical problem, it's a contract management problem.  Don't frig with the turnkey application.  It gives the vendor a get-out-of-jail-free card that allows them to ignore their service level agreement.

Either, 


Throw hardware at the problem (SSDs, bigger server, more RAM etc.), or
Talk to whomever manages the relationship with your vendor.  Push the responsibility for system performance back on to the vendor, and/or
Investigate the business case for moving to a competitor's product.


Messing with the application is the last thing you should do, particularly if relations with the vendor are getting tense.

You need to make sure your management knows that the performance issue is down to the architecture of the system.  Get that in writing somehow; write up a report describing the issue and the reasons you can't fix it unilaterally.  Make sure it's emailed to all of your internal stakeholders and you have a record of it being sent.  Arrange a meeting to follow that up and explain it.  That covers your arse - this is quite important if the vendor (or the party that authorised the purchase) tries to get political.

Now, escalate the issue with your I.T. management function, making sure it's well documented.  As you understand the root cause of the problem, this shouldn't be an issue.  The vendor has sold you a system that is not fit for purpose.  It's now a management issue.  Light the blue touch paper and stand back.

If you've already done this, then you've done everything that you can do without interfering with the application itself.  Again, I can't say this strongly enough: do not do anything to the application that could void the SLA.