OK, I sincerely understand your concerns. I work for a council creche service and I would have been given my marching orders for 5 cases where children have 'escaped' from premises. There is absolutely no excuse for lax child security.
However I am aware that you say the school is small and your daughter is new and young. From bitter experience I know how awful it is to be the child of the parent who is perceived to be a 'trouble causer'.
You have raised the matter with the Head and the govenors have agreed to raise it at the next meeting. I would also be tempted to take the matter up as a health and saftey issue with the Local Education Authority
This should be simple to fix. They need an extra handle high up. Ds1's special school has these on all the doors (so you open the doors with 2 hands, one handle at normal height, one at the top of the door). Okay ds1 can open them both now he's 9 but it worked when he was 5. Dead cheap too.
I agree this is serious and needs to be addressed now, not after endless meetings.
I am (as penance for my sins in a previous life) a school governor.
It is very important that you get a copy of your school's complaints policy - it may even be on their website - and then follow it. If you do need to take this beyond the level of the school/governing body, they will be looking first to see that you went through the school's complaints process (and with what result).
Where you go after the school/GB is, of course, up to you, but I don't see what you could gain by writing to (say) your MP. Your MP may be able to bring pressure to bear - if s/he chooses to take up your case - and you may gain some publicity that way, but the MP has no formal responsbility for your school or any other. LEA seems more likely if you need to escalate the complaint.