Since you're leasing, why don't you just fine a ready-made horse more suitable for your goals? If I was the owner of a leased horse, I'm not sure I'd be too keen on a strange trainer coming in and working with it.
Agreed. Unless the training of the mare is somehow worked into the lease agreement, I'd say the final say on whether a trainer should be brought in, and who that trainer should be, very much belongs to the owner. Unless this is something like a lease-to-own type of situation, in which things might be different?
That being said, are you thinking about hiring a trainer because you are encountering training issues beyond your ability? Or because "everyone has one?" The reason why trainers are recommended so highly on this BB is because a lot of people crop up who ask questions or have problems that honestly need more experienced eyes on site to truly help. Bearing in mind that I don't know you or the horse... if you feel confident to address the holes in the mare's training, or lay that foundation, I would tentatively say that you don't need
a trainer. If you are meeting one roadblock after another and having difficulties, then I'd recommend talking to the lessor and discussing what would be best for both of you in terms of either bringing in someone to help (whether the lessor, assuming that they are more experienced with green horses, or a mutually chosen/agreed upon trainer), or you moving on to a horse that is a better fit for you.
Putting myself into what I understand of your situation, I would go ahead on my own until I met a problem, and then discuss the next step with the lessor. But then, I like working with horses like you describe - comparatively older greenies with a few rides and a solid start but not a lot of polish. If she were untouched or had training problems, it would be a different story.
So, after that rambling... do YOU think that you need a trainer's help to teach the mare what she needs to learn for her future job as a reliable pleasure horse? Be honest with yourself, your abilities, and your expectations, and you'll make the right decision.