If it was me, I would decide based on what useful things you are getting out of this therapy, in general. Cost/benefit. Also, have you experienced similar obsession about other people (including other Ts) as well or is it only with this T? If only with her, maybe ending it is not a bad idea, even though it would quite likely be very hard in the beginning, much like quitting an addiction. But if this is a larger pattern, you might just find another person for it.

I am saying these things because I absolutely tend to do "research" like that about whoever interests me, just keep it to myself. The difference is that my curiosity usually doesn't last too long about one person and there is no frustration, because it is just curiosity that most often arises and dissipates pretty quickly. Of course your situation is different given how preoccupied you are with her emotionally. I think most people freak out at least a little if they learn someone is obsessed with them, unless they personally get a kick out of that type of attention, which is more an issue IMO... Just suggesting that a little freaking is probably the normal reaction, but I hope she will be more rational about it.

In any case, I would consider the overall benefits of the therapy when deciding whether this long-term preoccupation is worth it.