Going open source may be a solution too? Godot Engine is a self-contained 40mb executable (no installation) for Windows, Mac, and Linux, is easy to learn, (much) more powerful than either Gamemaker or Construct, and completely free. No hidden costs, and there's even a "kids can code" YouTube channel to help the students along.
And it's a lot of fun to learn and use. For example, Godot has a built-in animate everything you want animation editor, which is super fun and liberating in your game creation process: even cut-out animated 2d characters may be constructed and animated right in Godot.
On top of all this, your students have access to the full version at home or anywhere, really.
I mean, your Construct 2 access and use is going to end at some point in the near future, and Godot seems to have all the advantages you want in a classroom environment: free, student accessible, no installation required, great support and community, a wealth of online tutorials and resources, 2d and 3d, native exporters, web export, animation timeline, simple to learn Python-based GD script (and Python is proven to be a perfect introductory language for young learners).
The only drawback would be the lack of visual programming - although Godot does have a visual programming option, it's not that great compared to Construct's event sheets. However, it seems plans and ideas are going around to fix this in the future. On the other hand, your students would be introduced to Python syntax, and learn "proper" programming, and Python is a very popular language nowadays - so you'd be teaching them a real-world language too.
Anyway, just throwing it out there. Schools have small budgets to work with as it is. You may have to wait for Godot 3.1 (out later this Summer) for lower-spec Opengl v2 support to be implemented again.