All research on saliva tests I've found involve antibody detection which, as you suspected, can produce false negatives depending on the disease process or the presence of immune disfunction (such as due to AIDS). Thus, if you can isolate T. gondii gDNA from the saliva, a genetic assay may be more sensitive. However, Amato Neto et al. report:
In 26 patients with AIDS, including six with tomographical evidence suggesting cerebral abscess caused by Toxoplasma gondii, no parasites were found in the saliva by the use of the mouse inoculation model. This trial [suggests] that the search for Toxoplasma gondii in the saliva is not an alternative method for the diagnosis of toxoplasmosis in patients with AIDS.
So perhaps the most difficult aspect of generating a gene based saliva test is actually isolating the parasite from the saliva. I was unable to find other isolation attempts, but my research was very brief.
As for which genes to detect, studies involving human blood samples have found that PCR amplification of B1 paralogs provides a sensitive and selective method for detecting T. gondii. Burg et al found that there are about 35 copies of the B1 gene in T. gondii and also state that:
Because of its repetitive nature, the B1 gene is an attractive target for detecting T. gondii parasites through amplification of B1-specific DNA.
They were able to amplify and detect the gene from crude cell lysate as well as from purified DNA samples containing a 1:10 000 ratio of parasite to leukocyte. Furthermore, they could not detect the gene when assaying other organisms. Ho-Yen et al. later attempted PCR amplification of the same genes in human blood from patients with suspected toxoplasmosis and found that:
PCR can be reliably used on human blood samples. For a reference laboratory, this is a valuable addition to its array of diagnostic tests, and provides a more rapid assay than animal culture.
Jones et al. compared detection of the B1 genes with P30 and 18S rRNA genes in aqueous humour. They were able to detect 50 fg of gDNA using the B1 assay, which is equivalent to a single parasite. They concluded that:
The B1 PCR protocol appears to be the most sensitive protocol in the detection of T. gondii DNA and has been successful in identification of T. gondii DNA in ocular fluids and retinal sections.