I’d lean towards option 3. I view this site as an aggregator and I see plenty of broken images when scrollling through here. It’s not your job to host or proxy those images and working in infrastructure it lends to a lot lighter weight instance that’s easier for you to manage without having to worry about disk, or even worse, bandwidth costs.

The only way I would see option 2 being more beneficial is if it was truly a temporary cache solution where the image gets pulled in 1 time, served out to all lemm.ee users for a period of time (say 24 hours). This would reduce the chance that you end up rate limited, while allowing users to still see the image served via lemm.ee

The proxy on every request solution just seems poorly implemented to me for the reasons you say.