I had a Bendpak 2 post car lift installed in January. It is a superb piece of kit but totally unsuitable to use as a parking lift as would be any other 2 posters, they are not designed for that purpose.
As you already know headroom is your problem and with only 2.2 metres any commercially available parking lift will probably not work.
This one looks like the posts themselves are longer than 2 metres.https://sm-t.co.uk/product/bonar-parking-4-post-lift-manual-release/
If you garage is 3 cars long you could have a stack bespoke made and ramps to get to the upper level, it could be made as wide as the garage and double up as a service lift.
Just a thought, can you raise the roof? (don't mean on a Friday night at the local