Am in the might not be the right barn for you camp and the fact trainer is looking for something young enough to bring along is a yellow flag. That is not what you need to learn the HJ show ropes. You need to lease an older BTDT schoolmaster type and you do not need fancy.
Trainers often push less finished, fancier horses on clients to create additional income for themselves via training, lesson and pro ride costs. No blame there, it is their job but is not what you need when you are looking for an entry level lease horse. Smart, savvy trainer will pair you with what you need now and help you as you move forward which will insure more modest but longer term income for trainer and boost for their reputation. You need to find one of those.
The great thing about lease horses is you can walk away at the end of the term if you grow out of the horse ability wise, it has soundness issues and/or you realize you are with the wrong trainer at the wrong barn.
Seems to me that your budget should allow you to sample several different barns/programs and half or part lease suitable horses as you advance. Cannot recommend a year lease for you right now in your current situation, think you can do better as you get more familiar with what your options are and who/what is out there barn and trainer wise.
Can you just go watch a show that you are considering showing at? Sit and watch carefully, hang around the schooling ring and in gate, watch how trainers prep their clients and debrief riders after the round. It can be very revealing and costs nothing but your time.
Often the best are not the best known and do not openly advertise.