It sounds to me like the OP has a great situation in many ways, and is just wanting to vent a little and get ideas about how to do what she wants without leaving her barn. 
OP: I get your feeling of being a square peg! I’m in an eventing barn and getting my trainer to go to jumper shows is really hard because a) she is eventing and/or coaching at events from spring to fall, and b) H/J shows don’t have ride times so it’s often a matter of spending an entire day with one client sitting around hoping the ring conflicts won’t be too bad. It’s just not worth it unless she can meet me, help me out, and zip back home to do the barn and teach, and often the time window is so freaking wide that isn’t possible. I work with a couple of trainers but they are both eventers, so it’s basically the same situation. I go to Aiken in the spring for a few weeks and work and show with a H/J guy down there who is amazing, but that’s just a small slice of the year,
My trainer takes her own horses and brings her other students to jumper shows in the winter and we do get to some shows during eventing season, but it’s always difficult to make it all work. Doing rated shows is especially hard because they are too expensive and not as attractive for eventers who just want to practice stadium without paying stupid amounts.
This is my choice and the benefits of being at this barn and with event trainers very much outweigh the costs, but I’ve had to be really self-sufficient to be able to compete. I have my own truck and trailer. My horse will self-load, and he’ll stand tied to the trailer or on the trailer without fussing by himself all day long. If it’s a multiday show I generally hook up with some H/J trainers I know if my own people can’t be there. I do my stall and such to save money but I get to be with the group and have some company.
And, perhaps most importantly, when I am competing regularly I am fully capable of showing without a trainer. If I’ve been out of the ring for a while for whatever reason I usually need someone to hold my hand for a show or two, but a huge goal of mine is always to be able to show with just a friend helping on the ground so I don’t have to depend on fitting myself into my trainers’ schedules.
All of this takes some resources (having my own rig is not cheap!) and a lot of confidence (thus the occasional need for hand holding), but it’s worth it to be able to do what I want with my jumper and continue to ride and train and board with my excellent eventing friends.
Hopefully you can figure out something similar, OP! It is possible to do this, but it can be tough to find the right path. Good luck!