Global shortcuts (Cmd+Space, etc) disabled by physical mouse?!

I have been trying to use a keyboard and mouse with my iPad but I keep hitting a number of infuriating corner cases that makes keyboard input unreliable.
Intermittently the global shortcuts stop working but only when the mouse is plugged into the hub. These include Cmd+Space, Cmd+Tab, Cmd+H, etc. Unplugging the mouse and plugging it back in sometimes fixes the issue.
There is another potentially related issue that isn’t what is described in the title:
– The cmd key sometimes gets “stuck” between apps. The common issue is Cmd+Tab to safari, read something, Cmd+Tab to Gmail and start next paragraph and press enter but cmd is stuck so it sends my email!
– Capslock will frequently turn on when switching between apps.
Is there some setting I am messing up somewhere?
The Setup
– iPad Pro 11” 2018
– iPadOS 13.2
– Apple USBC HDMI hub
– Logitech M570
– Logitech K380
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