I think I have a solution that doesn't require any permanent mods to the Clarity, and in fact might work for any vehicle. It requires use of two "Add-a-Fuse" devices, and knowledge of which always-on fuse location feeds Homelink and another fuse location that's on an ignition-switched power feed. In an ideal world, you'll find an empty position in the fuse panel that has ignition-switched power going to it. I've identified the Add-a Fuse devices as (1) and (2).
Verify that the vehicle is "off".
Remove the fuse in the fuse panel that feeds Homelink. In one Add-a-Fuse (1) device, leave the primary fuse location EMPTY and place the removed Homelink fuse just removed in the Add-a-Fuse (1) added circuit fuse location. Plug this Add-a-Fuse (1) device into the Homelink fuse panel location, but plug it in BACKWARDS. The dangly wire on this Add-a-Fuse (1) now goes to the side of the fuse location that feeds the Homelink circuit. Measure the voltage on the dangly wire of the Add-a-Fuse (1) to be sure. If you find 12V there, it's not plugged in backwards...flip it around and measure it again. It should measure as 0V.
If you're following along here, you may have figured out what I intend to do...
Remove the fuse on the identified ignition-switched circuit (if it's not an empty location). In the other Add-a-Fuse (2) device, leave the primary fuse location empty (for now). In the added fuse location of this Add-a-Fuse (2) device, place a fuse in there that's the same value as the one removed from the Homelink-feeding circuit. Plug this Add-a-Fuse (2) device in the ignition-switched circuit location. Measure voltage on the dangly wire. It should be zero.
Verify that neither dangly wire is touching anything, and turn the vehicle "on"
At this point, the Homelink and ignition-switched circuit identified should still be dead, and it's possible that the vehicle will throw a code or two. Somewhat expected...
Measure voltage on Add-a-Fuse (2)'s dangly wire. It should be 12V. If it's not, turn off the vehicle, flip Add-a-Fuse (2) around and turn the vehicle back on. Again measure voltage on Add-a-Fuse (2)'s dangly wire. It should now be 12V.
Turn off the vehicle.
If you are using an ignition-switched location in the fuse panel that had a fuse in it, do the following. Otherwise skip down to the dashed lines.
Remove Add-a-Fuse (2) and because you took a fuse out of that position, place that fuse in the primary fuse position of Add-a-fuse (2). Plug Add-a-Fuse (2) back in the location you just took it out of, with the same orientation it just had.
What you now have is a dangly wire from Add-a-Fuse (1) that is the feed to the Homelink circuit, and a dangly wire from Add-a-Fuse (2) that only has 12V when the ignition is "on". If you tie these two dangly wires together, the Homelink circuit will only have power when the ignition is "on". So...
Cut off the crimp connector from one of the Add-a-Fuse devices. (It doesn't matter which one.) Strip that wire, and crimp it to the dangly wire attached to the other Add-a-Fuse device.
You need to know which circuit fuse location goes to Homelink.
You need to know which fuse locations have ignition-switched power going to them.
You need to know which other devices in the vehicle are fed from the same circuit as Homelink,
and how they would respond to intermittently cutting power. (Quite important.)
Screwing this up could potentially cause severe harm to the vehicle electrical system. (You could brick your vehicle.)
Fundamentally, I think this could work, but I haven't tried it...yet.
CANBUS is a HUGE potential monkey wrench.
This requires that one knows the basics of how vehicle electrical wiring works.
Having access to the vehicle shop manuals, especially for its electrical systems is practically a requirement.
This is not a step-by-step guide to be used by anybody not already comfortable working on vehicle electrical systems, and is in no way an encouragement to try this. I intended this to be more of a proof of concept post.
Doing any of this is entirely at your own risk, and I personally bear no responsibility if you try, and things go sideways.
Please comment, even if it's to say that this is nuts (but please tell me why), because I know that this is thinking waaaaay outside the box.