10 Reasons Why The New Doctor May Actually Be River Song
8. Genetics Of A Time Lord
If the Chameleon Circuit is not enough to persuade you, then perhaps their genetics will. River Song's genetics are of particular interest. While she is the child of Amy Pond and Rory Williams, she is also a "child of the TARDIS", created in the Time Vortex. She is, by her own right, a Time Lord.
While this does not make her the same person as the Doctor, it does mean that she would register as a Time Lord on the Judoon scanner - just as she had previously scanned as "human" under the Chameleon Circuit.
As Gat says, "two of the same Time Lord can't occupy the same space at the same time." Therefore, it should be impossible, or certainly dangerous, for both Whittaker and Martin to be on the ship together. However, there is no visible complications. Perhaps as one of them is not who they say they are, they are not the Doctor.