It would be polite to send an email to the Primary Head to say you are considering another school and that they will be seeking a reference.....it wouldn't be great if the first the Head knew about you considering elsewhere was a request for reference from the other school.
As a governor, depending on how well you know the Head, it might be polite to have a conversation about it too - it doesn't matter if you find it a little awkward - if you knew a Boss well and were applying for another job, it would be a courtesy to let them know and briefly talk it through - this isn't asking permission, just telling them. The Head will be impnterested to know why you're doing this - any email or conversation can be worded to show you aren't letting them know so you can debate it, but to let them know and give reasons.
And it's worth mentioning positives about the current school. Even if mentioning funding cuts as being a worry, be clear your DC has been happy and that in many ways, it's regrettable. Being positive is important.
Don't be worried about this. the Head is a professional. Every day, they deal with parents making choices which might not be those they would make, and they deal with it. Just be honest and clear....not apologetic or grovelly, but friendly, clear and show appreciation for what the school has done for your child and for them writing a reference.
And if you plan to continue as governor, make that clear too.