If this is the same thing as a rapid-up jack, I have not been pleased with them. Under normal use, the stress on the handle to quickly lift the car causes them to break at the hinge position for the handle. A jack purchased from Harbor Freight was used. Thanks to my new two-speed Sears jack, I can quickly raise or lower heavy objects, and it returns to its usual speed after reaching its weight capacity.  

So does a 10" wheeled mini fit on a QuickJack 3500? The yellow car pictured above seems to be making it work.

My two car garage.  I cut and raised the horizontal roof supports and reattached them higher up to allow the Mini (the tallest of my cars) to fit both above and under the lift when it's up.  Four post lift was about $1900 several years ago.  Has been a blessing for and old fart like me, as I can get under the cars on my roller stool with ease.

if you mean the type of quick jack the formula cars use at the race track, like the B-G type, the severely lowered race Mini offers a significant challenge to those. I have an extended T handle for mine and I can raise the rear, but not the front.  I only mass about 150lb, maybe if I had a few friends pulling with me we could raise the front.

 

I bought a mid-rise lift used for $800 usd.

That's an entirely different thing, I am also interested in opinions on this.

I wanted to put a 2 or 4 post car lift in my garage but I don’t have the ceiling for it.

The one I was looking at were these ones:
https://www.quickjacklift.ca

  
here is one of the few pics i found online

I guess if you've been looking at them, you see the benefit of them for your application(s) and can warrant the cost and space involved with having them.
Never used them, but would have a question of whether they're long enough to reach the load points under the Mini to properly support it. There's no side or middle support under the Mini. The main support is at the 'bulkhead' areas, front and back just before both subframes. It would have to touch both of those end-areas to not only be effective, but safe, and non-damaging. 
If words aren't painting enough of a picture, watch this video to show you what I mean about jack points. https://www.youtube.com/watch?v=IXSU2Oxh6MU

If this is the same as a fast up jack, I have not been impressed, using them in daily life the loading on the handle to quickly raise the vehicle makes them fail at hinge point for the handle. this was of course a harbor freight jack. I now have a two speed sears jack which is nice, fast up to the load, then normal from there.

Not sure my interpretation of a "quick jack" is the same as yours.... but the Works used what I would call quickjacks on the rally cars back in the day.

Not sure what you would hope to gain when using one unless seconds matter with your tyre changes...

They would seem to take up a lot of space..

Cheers, Ian

Has any used a quickjack with their mini?
Been looking at them and was looking for feedback.