It seems that your concerns for your daughter can be handled in what you have already shared. I believe your comment "When I tried to go in with Ellie it was made clear I was within my rights but unwelcome, " may be a guide.
-- I think we should be wherever we feel we need to be to protect our children-- just because others may not feel welcoming- would not be enough reason to prevent my presense. Yet I would do all I could to make my presense be as positive as possible!
If especially-- you can trust God that it is ok for you to attend your daughters class-- and do it with a Christ like attitude-- so you are thankful that there are sisters who are willing to take the time to teach the class , and on the day you teach Relief Society-- you could either bring her with you, or see if you pray about it-- if Heavenly Father would watch over her in her class- or ask another friend to attend with her.
You might even acknowledge that you *might* be just a bit over protective,
but you would rather err on the side of prevention than repent in sorrow later. You and your hubby are (on earth) the ultimate persons responsible for her care-- and I'd go with your "gut" and not have her in there if you don't feel good about it. --
the good news is that it is just a few months away from her graduating to a different class/teacher.
Part of the possible damage is though, that it is sad for her to feel "unsafe" in church
but it is worse to have her have a bad experience and then totally believe she is unsafe! I would call the "naughty chair" a bad experience already.
I didn't see an age given for her. But I think we don't assign negative traits, nor have time out more than one min. for each age OVER 2 years.
(I do NOT think a "naughty chair" is anything the church has EVER used in my 50+ years of teaching primary-- as I believe it gives a "negative" label to a child--
the "love and logic" principles (there is a web site) work I believe like Heavenly Father works with us.
It may be that this trial/challenge will help you to grow as you can
a. show you will be with your daughter- though you suffer a TINY bit feeling unwelcome
b. help you practice being Christlike
c.

get you a calling to be her teacher?
d. be a good example to others
May God bless you and your family and us ALL to grow more Christlike. Gramajane