Crash course in how public education works:
You either own a home or live in a dwelling that is owned by a landlord, so that means that property taxes are being paid. Part of those property taxes are used to fund the public education in the district. You (or your landlord) do not pay property taxes in the other district, thus you are not entitled to the benefit of the education that is being offered in that jurisdiction.
Since it is an elementary school, you likely would want to contact the principal first. It all depends on the administration structure of your district - there might be someone who is designated to handle issues of residency and transfer. The principal should be able to direct you to the appropriate party who handles these inquiries.
Some people are able to get their student transferred to an out-of-district school because they were 'accepted' by the other district due to the fact that the family paid tuition (usually the per pupil spending expenditure...could be anywhere from $3500 to upwards of $12,000). Even if you could afford to pay the tuition to the new school (because, since you don't live in the district you are not paying taxes to the district), the school typically does not have any legal obligation to accept your student.
Quite honestly, if you really love this other school your best bet would be to move into the district so your daughter is legally entitled to receive an education that your property taxes are funding.