Since you are asking about renting a car hoist, I have to ask, what do you think one would cost to rent?
About ~5 years ago my brother bought a "mobile" car hoist from car quest (not the cheapest place in town but he knew a guy and I think also got employee pricing) for around $1200 for my dad. Best investment ever. (Likely more now though)
Now, we took it home in a 1/2 ton pickup (it was likely overloaded) with no freeway driving. I think it weight around 1500lbs. Car quest put it in the pickup with a forklift arm type thing. We dropped (yes dropped) it out of the back of the pickup with all toes clear of the landing. (It was fine.)
It is a floor hoist that swings up kind of like a motorcycle lift (but this is either to the front or back of the car). Kind of tippy if you don't lift high enough (full to most lift puts the raised part square over the base). Never had a problem with the tippy part though. (Which could also be part of a non-level floor problem.)
"Mobile" I don't know what they really meant by this, other than by you "can" move it. It has 2 wheels on one end and the compressor part is on a stand that can "lift" up the other end to make it move-able. Not intended for moving between uses, but just so that you "can" move it. (We used it for moving between uses, lots of work, in the winter and it found a home under a car in the summer. Rustbelt, so didn't want snow/salt on it.)
Very happy with it, but you can't drop a gas tank with it unless it is a trunk tank (and than it likely doesn't "drop" anyway).
Lastly, the delivery cost to rent such an item (unless you can haul it and LOAD it from your house) would likely be better spent toward the purchase of the hoist.
Hope this helps. (Feel free to ask question about what we got for my dad, I don't have pictures, but will look on the internet for some.)