Short term, low effort suggestion. Get a pack of very thick socks. Put them on the bike's pedals, handlebar ends/ brake levers. And something you can wrap around the chainring, or get a chainring protector. The socks will prevent the bike pokey bits from scratching your car
Mid term, medium effort, medium cost DIY suggestion: a semi permanent over-liner for the interior of the booth.
At least here in my country I can easily obtain a carpet material that has some cushion between a fabric like backing and a vinil top. The fabric backing would not scratch the plastic panels of the car, and the vinyl top will be easy to wipe to clean the mud off. It is about 5mm or 3/16 thick in total.
With that or something similar I'd make a booth liner, so to speak. The idea is to fashion carpet panels conformed to the sides of your car. Usually the back of the seats and the inside of the rear door is flat, so that's the easy part. The sides however have a lot of funky contours. There you'd have to be creative with cutting and gluing or stitching, so you get at least a somewhat aesthetic side panel cover. You don't need to conform to every nook and cranny, just the general contour. The smaller spaces formed between the car's panels and the custom one could be useful to store/hide some tool or accessory.
Now find a way to attach it semi permanently. Your car may have small hooks or loops to secure cargo, those may be good enough if they are placed correctly. Another option, if the side panels are held in place with bolts or screws, is to use longer ones that could hold in place both the car's inner panels and your custom liner. I wouldn't recommend adhesives as the stronger ones may damage the car, and the others may be not strong enough or will not last attached long enough to be worth.
With a long strip of the carpet material, elaborate a liner for the rear bumper / trunk lip, and wrap it. You could fix it in place with zip ties or paracord.
If done aesthetically enough and the right material is chosen, it would give your car a "rugged" appearance, and if fit correctly, you don't have to install and remove it every time you need to transport your bike, bout it would not be bothersome when using the car for something else.
However, I really think this is a lot of work and would work more towards the value preservation of the car than to the current aesthetics. I definitely think that the way to go for the long term is the hitch receiver with a proper rack. I know you think $500 is too much for a single bike, but in your case, it is not $500 for a single bike, it is $500 for the aesthetics and value preservation of your car, plus the ease of use for you. that is, put "value" on the time and effort you are going to save on each an every trip you make with your bike in the car.