Although you are looking for non-Suzuki showrooms, but I have a recommendation for Suzuki showroom for maintenance. You can check out Suzuki showroom in front of Ganga Ram hospital. I don't remember the name of showroom exactly, but the head mechanic there is Adnan bhai. He is an expert.
For parts, you can contact Ittehad Autos or Rafiq Autos at McLeod road apart from Suzuki showrooms.