If you want to maximize your chances of getting the school eval to include the areas you want assessed, name them in your letter requesting evaluation. E.g.,
"I request special education evaluation for my child. I have concerns with his function in the following areas:" and then name the concerns, not the suspected diagnosis. The way a good school eval flows is not unlike what a good neuropsych does. No psych should be fishing for disabilities (although, unfortunately, I know some who do). He/She should be starting from the presenting problem, and then tailoring the assessment to the needs. If stuff comes up in testing, then you follow it up with further assessment. A comprehensive cognitive and achievement make sense as a starting point for school-age children with school-related referral questions, because it gives you a context, and usually tips you off to areas to pursue further, like working memory, or visual-motor integration. I usually survey any child with a vague referral question with a social-emotional-behavioral-attentional rating scale, such as the BASC-2, just in case there is something there that no one has mentioned. If I get a red flag, then I follow it up with more assessment appropriate to that area.
The number one factor is the specific evaluator with whom you are working, not what their nominal specialization is. That's why it's so important to talk with the evaluator, whether school or private, ask as many questions as you can, and give them a comprehensive view of your concerns and your child's background. Most school-based psychs would love to run an eval as extensively as private psychs do, but don't have the time to do so, except when forced to take the time by parental pressure or administrative pressure (which causes admin to cede the time). If you take the initiative to talk to them collaboratively and collegially, and subtly offer them the opportunity to exercise their assessment skills the way they were trained to, they are likely to do so. Certainly all the ones I know would love to have a case where they get to do more than a WISC and a WIAT and call it a day.
