The short answer to your question "Could Lee have lost on purpose" is no. There is enough substantive evidence that Lee was not a secessionist and was loyal to the United States until his state of Virginia seceded. But once that happened, Lee's loyalty to his state, which had a long aristocratic pedigree, took center stage, and he wholeheartedly supported Virginia and its decision to join the Confederacy. From a military point of view, Lee was known as an audacious and bold commander who was responsible for the Confederacy's long string of victories in the east. The fact that for almost three years he successfully parried almost all of the Army of the Potomac's attempts to defeat him is more than enough confirmation to refute the proposition that he lost on purpose.
I must add, however, that the question makes for an interesting debating point.