I'll pipe in, like LMom, from a native language perspective and not immersion. I am a FIRM believer in the opportunity to grow up bilingual, biliterate, bicultural... or any of the above.
I think it's awesome that you guys have the opportunity to do these immersion programs - if i could i would absolutely do one for my boys (in a third language, since we speak two at home).
I do find it incredibly odd that they don't encourage (or, really, allow!) you to teach the skills at home in english. I think to be truly bilingual learning them both simultaneously is really the way to go. It's how those of us who are fortunate enough to grow up learning two languages at home generally do it...
We have been speaking mostly spanish at home from day one to the boys (i say mostly b/c while I'm a native speaker and my husband speaks it fluently, we have a bad habit of speaking to eachother in english so the boys are exposed to it all the time. Also, if there is any non-spanish speaking friend or relative around, we'll do english or a combo).
They are now both pretty much bi-lingual (at 2 and 6 they understand everything in both languages and...when pressed will speak spanish quite well - though they do get lazy...). Big man started reading just shy of 4 , in english. A few days after i realized he was reading english, i brought out one of our spanish books and asked him to read... he did it fine. He taught himself to read, and essentially did it in both languages simultaneously - because he was exposed to both in the same manner - books at home, music, talking - all the natural, daily activities of life. Yes, he's better at english, but we don't have a lot of child-appropriate chapter books in spanish, so that doesn't help...
If your daughter is being exposed to the "normal" activities in spanish at school, being exposed to them in english at home should, I believe (and agree with what's been said), only enhance her learning at school and make it all "click" a little easier. Not to mention that successes at home (hey, i can DO it!) will make hardship at school a little bit easier to take AND, i think, will help her realize she CAN do the spanish too.
The exposure to different languages helps on so many levels, i would hate to see her frustration at this point make her lose out on the many benefits of being bilingual. It's NO fun to see your child frustrated at school - but i think with gifted children, frustration at being too challenged vs. not challenged enough sounds a bit refreshing... As Texas Summer said - you know your daughter best and will utlimately have to make the decision that makes you all as a family comfortable. But i'd encourage her to give it a little longer, try english at home, and see if that helps!
Buena Suerte! ;-)