I don't know anything about this JUCE library that is used for the Axe-Edit UI, but I found this post about their renderer:
GL image rendering improvement | JUCE C++ Library
The claim is that their GL renderer runs ~250x faster than the CG renderer by caching images on the GPU. Sure enough, the time spent in Axe-Edit is in the CG renderer drawing images:
+ ! 340 juce::Graphics::drawImage(juce::Image const&, int, int, int, int, int, int, int, int, bool) const (in Axe-Edit) + 232 [0x4cd52]
+ ! 340 juce::Graphics::drawImageTransformed(juce::Image const&, juce::AffineTransform const&, bool) const (in Axe-Edit) + 109 [0x4cb5d]
+ ! 340 juce::CoreGraphicsContext::drawImage(juce::Image const&, juce::AffineTransform const&) (in Axe-Edit) + 39 [0x6154b]
+ ! 340 juce::CoreGraphicsContext::drawImage(juce::Image const&, juce::AffineTransform const&, bool) (in Axe-Edit) + 1120 [0x61226]
+ ! 340 CGContextDrawImage (in CoreGraphics) + 429 [0x924ab30c]
Anyway, hopefully one of the Axe-Edit engineers will take a look at this info.
This performance lag is not just the text window, although it is more pronounced there. I see similar FPS speed limits when dragging things in the layout view. For example, drag one of columns around (so you are dragging all four blocks) and it will not keep up with the mouse pointer.