As a teacher, you can refer a child for special education services at any time during the school year. When the referral has been addressed, the student(s) will be tested by the school diagnostician or other professional designated for testing students.

Hello,
 
I work for a large school system.  My experience is that teachers can refer students at any time of the year.
 
Do you have access to reading specialists, early education co-ordinators, speech and language pathologists, early intervention workers, psychologists, physiotherapists, and occupational therapists?  We have teams of people who can assess kindergarten students throughout the school year.  
 
If you already have special education assistants, could some of their time be devoted to working with these three students that you've mentioned, after an assessment of their needs has been completed?
 
I haven't worked with kindergarten students directly, but I have spent many years working with students from grade one through twelve.  I've provided intervention throughout the school term, and significant progress can be made if the interventions are targeted to student's specific needs.
 
Perhaps your school system could re-consider how and when they allocate resources throughout the school term.
 
Sylvia HS
Reading Specialist, Author
 
p.s. You might look at the website of the North Vancouver School District 44.  They have routinely screened kindergarten children in January and then provided intervention during the spring term, and their district results have been phenomenal.

Students should be screened a few times a year.  I think screening in the Fall is fine - but also very important in the Spring.  If a student starts the IEP process in the Spring they can get started in September versus having to waste another Fall getting the help that they need.