My recollection of the parking sensors is that they have on the order of a meter of minimum distance, but that will depend on how electrically isolated you can make your transmit and receive circuits and mechanically isolated your transmit and receive transducer mounting.
If you’re going to count on ultrasound, you probably need to price the initial effort to include the cost of a maxbotics module which is advertised as working down to 20 cm, maybe you can eventually come up with something cheaper. If development goes smoothly you might break even before the equivalent of 100 maxbotics modules, but if you run into difficulty you might not.
I have to wonder if you couldn’t do something with transverse beams IR beams.
Or design the bin with a door to keep out critters, and put a sensor on that so you know when it’s no longer closing because the bin is overflowing. If someone jams something in there and it’s not actually full, someone else will probably push it in (or let it fall out) while adding their refuse so a lastingly stuck door is likely a useful indication if the bin needs service, though sadly not one of if it may get through the coming day without needing it.
One solution actually on the market (“bigbelly”) is a solar powered compactor (obviously with a battery to store up the energy for brief operation) presumably measurement is done by monitoring the position at which the compactor stalls out.
While you’re at it, replace the Arduino with something more suitable for LoRaWAN. Even if you want to stay Arduino style, get something with an ARM Cortex core.