I don't believe that Descartes answered this particular question, but if I had to answer for him I might argue the following:
- Doubting by definition is experiencing.
- Whoever doubts is experiencing something.
- Let's assume that Descartes is doubting according to God's decree.
- Let's also assume that the doubt belongs to someone else.
- Since Descartes is doubting, he is experiencing something.
- In this case, he is experiencing something that belongs to someone else.
- If Descartes is experiencing something belonging to another, he is experiencing.
- He experienced, therefore he was.
That's a roundabout way of saying that neither the cause of the doubt nor its belonging to someone else (whatever that might mean) is relevant, given that you are assuming that he is experiencing doubt. As you said, "But the only thing he didn't doubt was his doubting."
Philip Klöcking suggested that this question might be addressed according to the fact that Descartes rejected the idea that God is a deceiver:
"From this it is manifest that [God] cannot be a deceiver, since the
light of nature teaches us that fraud and deception necessarily
proceed from some defect." (First Meditation)
However, that doesn't rule out the possibility of deception coming from another source, which was an idea that Descartes also entertained:
"I shall remain obstinately attached to this idea, and if by this
means it is not in my power to arrive at the knowledge of any truth, I
may at least do what is in my power [i.e. suspend my judgment], and
with firm purpose avoid giving credence to any false thing, or being
imposed upon by this arch deceiver, however powerful and deceptive he
may be." (First Meditation)
In either case, he would be doubting.