Protein precipitation does give you ion suppression, and it has little to do with insufficient precipitation of the proteins. It has to do with ion suppression due to a large number of coeluting compound, that actually may vary from sample to sample.
To get around ion suppression, you have several choices. You may be able to get around it by using another LC/MS method (e.g. HILIC, if suitable). You may use solid-phase extraction or liquid-liquid extraction. Among solid phase extraction, you have several methods available. A simple reversed-phase method, a more complex pH-solvent reversed-phase method, and a simple reversed-phase ion-exchange method on a mixed-mode sorbent. I have listed them in the order of increasing sample cleanliness, i.e. less ion suppression.
I can send you an article on this.