Unfortunately the microphone in the remote is only listening when the button is pressed down and at this time there is no way to have it set to be actively listening all the time. I'm not sure if this would help in this situation but the numeric keypad works as a predictive search feature, so if he's able to press those buttons he may be able to search that way. You can find more information about using the remote on BlueSky TV Remote Control and we also have information on the accessibility features on https://community.shaw.ca/docs/DOC-9924
I'm no expert (I probably couldn't make this myself) but it sounds like you could use an Off-delay timer also called a NOTO (Normally Open Timed Open) relay.
It's a bit of electronics that will leave a switch turned on for an adjustable period. So the person could momentarily push a new separate button (with their mouth stick) and it would then turn the mic button on for them, but when he releases the new button the mic button would remain 'pushed' (on) for a little while on its own. Setting the delay adjustment to something like 5 seconds would give the person time to voice a command before the mic turns off. You could also add an On-delay timer if the person needs a few seconds to get the stick out of their mouth first before the mic turns on.
The electronics are not expensive (under $10) but this is a custom application so you'll need an electronics technician to make it and wire it to the mic button on the remote for you. I figure it should cost less than $50 total though.
Edit: something like this one from Ebay would probably be a good starting point (US $5).