Find another barn to ride at, period!
If the barn thinks so little of their students comfort, safety and learning correctly to not have multiple size appropriate and safe saddles for their mounts and students to choose from... do you seriously want to ride their?
I would not!
You as a student should not be purchasing a saddle to ride a lesson horse. In fact, many barns will not allow you to use your own personal saddles on their horses to ensure correct fit, safe tree and your own safety being foremost on their agenda.
Even school-horse lease horses usually have tack supplied for the same reasons...