Since my daughter is very shy, I stayed to observe her for a couple of hours on her first day of preschool (Montessori). The first thing the teacher did after introducing herself was to invite my DD to color at the table. While my daughter (not quite 3 at the time) was coloring, the teacher reached over and drew a 7 on my daughter's paper, and asked her if she knew what it was. My daughter told her it was a seven. The teacher grabbed some beads and some flash cards, and asked my daughter to count the beads and indicate the matching flash card (1-9). DD made a couple of counting errors (she's used to being able to move items individually as she counts them, while the beads are stuck together). If she hadn't made those errors, I assume the teacher would have tested her on addition/subtraction (where she would have failed all but the simplest test). After counting, the teacher checked to see if my daughter knew 0 and 10 from the flashcards.
At any rate, it literally took less than 30 minutes for the teacher to zero-in on my daughter's numeracy level in a low-stress, and unassuming way. I decided I didn't even need to give the teacher the document I had written detailing my daughter's abilities. She just didn't need it.
DD has been there less than a month now, and she told me today that she wishes we had the easy readers her school has. Apparently she's been reading things slightly more advanced at school than what we have at home (we're going to the library soon to remedy this.)
I wasn't there to witness how they discovered my daughter's literacy level, but it seems they figured it out within the first few weeks (possibly within the first few days).
My daughter doesn't have any 2E considerations, but she is shy, and she does sometimes amuse herself by purposely giving wrong answers.
If the teacher was unable to correctly assess my daughter's abilities after months, I would definitely consider taking DD out of that class. Any learning that happens with a teacher that doesn't know the student's abilities is just dumb luck, and that isn't worth paying a premium for (I have cheaper alternatives, but send my daughter to this school because I expect them to facilitate her pace of learning).
I would definitely investigate your alternatives.