A/C relay bypassed with jumper, compressor turned on but clutch quickly got hot and smelly

To me, this sounds like your clutch, which is lucky because that's a (pretty) easy thing to replace (you don't generally have to replace the compressor at the same time, and therefore won't have to drain the system). That said, your entire setup does sound pretty hacky, and I'd suggest fixing it the right way.

First, having a jumper for the system should not be needed. The point of the electronics is to turn off the compressor if the system is in an unsafe state. If there's not enough refrigerant, it'll turn itself off to protect the compressor from getting dry. If there's too much refrigerant, it'll turn itself off to protect the clutch from trying too hard. And if it's too cold, it'll turn the clutch off to protect the evaporator from freezing.

Here's my guide on recharging A/C systems, which might be worth a read, even though it looks like you're already familiar with a lot of this. If you have to add refrigerant that often (once a year), you definitely have a leak. If you'd just like to replace the clutch and see if things work themselves out, that might fix your problem for now, but generally old O-Rings are your problem. The connection between the compressor and the pipes generally has one of the few o-rings your A/C system uses, and it's a common culprit. After 5-10 years, those can start getting old, and by 10-15 years, it's good to replace them. You do have to drain the system to do that though.