Please check the following website: http://www.gtfrealty.com/
This is the realty company we rent from and they've been great to us. Their search page has the option of selecting the school district. I'm not sure if they currently have what you're looking for but if you can keep checking back something may come up. There is currently a 4 bdrm listed that is more than the 1K you would like but it looks pretty nice for the price.
Best of luck to you and your family. I hope this helps give you one more option at least.
After looking on the MLS, there was only one listing that fits your criteria. It's available for immediate move-in. It's 3 bdrms-2 bths w/ 1,453 SF. The landlord is asking $1,195/mth. Please let me know if this works for you. I will be happy to email you the listing and schedule a time for you to view the property. I look forward to hearing from you!
Make it a Great Day!
Tamika A. Goree
Broker | Owner
The DFW Property Shop
http://www.trulia.com (Rentals Section)
http://www.zillow.com (Rentals Section)
http://www.realtor.com (Rentals Section)
Between all of those, you will get a very clear picture if you can find what you are looking for. Good luck!