I have no experience with immersion classes but I do have experience with reading in two languages. My kids are bilingual and I too think that reading carried from one language to the other.
DS5 learned to read in both languages at the same time. He did better with English than the other language, which btw is a phonetic languages. DS4 learned to read English first, then almost a year later asked to be taught reading in the other language. Long story short it turned out that he was already reading. He didn't need any lessons, he figured it out on his own. I am sure the fact that he could read in English helped enormously.
Go ahead and teach your daughter how to read in English, it may end up helping her with her Spanish.
