I Do Think The Local Shop Has To Recheck The Timing Belt and associated components, If They Haven’t Done That.
Taking you at your word, You paid them for a timing belt replacement as preventive maintenance on a car that was running just fine, put up with an improperly running car for several months, while giving them $400 more to make the car run worse and worse.
Have you suggested that the timing belt installation may have been faulty? Have they suggested it? Have they rechecked the installation or the engine timing, that would be altered by a botched job? What are they agreeing to, refusing to do, or saying about the matter? Are they just shrugging their shoulders, out of ideas?
If so, you may have to go to another shop for a proper diagnosis. Should it turn out that shop #1 botched the job, they shoud either fix it or refund your money.