Odor as a result of reversing

Does it smell like rotten eggs? If so, that's sulfur in the catalytic converter. The catalytic converter is getting clogged and creating the smell, and you're backing right into the source of the smell (the tailpipe).

A catalytic converter that's just starting to get clogged up might not always make a noticeable odor. I'd go ahead and check it. There's a cheap, but very backyard/backwoods way to check it. Basically, you buy a can of Seafoam from the local parts shop. Raise the hood. Disconnect the brake booster vacuum hose from the booster itself. Start the car. Open the bottle of Seafoam. Stick the booster hose in the bottle and let the engine suck the stuff in. As soon as the bottle is empty, shut the car off. Let it sit for at least twenty minutes (I suggest a couple of hours). Then take the car for a ride and drive it -almost- like it's stolen. Open the throttle up. Tip - this creates a lot of smoke. I mean a lot. So do it when the neighbors aren't around. And when you do go for the drive, get out of your neighborhood as fast as possible. If you can do it out in the countryside in an area where no one is around, that's even better.

This trick has been used to temporarily make a catalytic converter perform better. A few years ago, my wife had a 2000 Jeep Grand Cherokee, with the 4.0L engine. The service engine light would come on and the code would be a p0420, which basically translates to "get a new catalytic converter." I tried this crazy seeming trick and the light went off and stayed off for a few months. The truck ran better than ever, and even temporarily got a few extra miles per gallon. Of course the mpg's dropped back to their usual 16 mpg and the truck went back to running like it had 160,000 miles on it (smoothly, but not very powerful). After four or five months it came back on. I did it again and it worked again, and the thing even passed an emissions/smog inspection. We did this for two years and then sold the cursed thing. I'd have never done it but a buddy of mine suggested it, and he's one of those guys whose judgment I don't question when it comes to cars.

Here's a decent YouTube video: https://www.youtube.com/watch?v=1WWqkDo7KwY

Check it out. Give it a shot. Let me know what happens.

As for the transmission issue, I would seriously doubt that that's the problem. Generally, transmissions don't make smells like that. They just start slipping and get worse over time. Plus, the issue that you're talking about is with automatic transmissions, and yours is a manual transmission.

As a test, take the car out to a spot where you can back up close to a wall. Now with it out of gear, open the windows and just step on the gas a few times. Do you smell that same smell? If yes then it is a failing converter, the extra fuel when you step on the pedal is enough to overwhelm it and cause the sulfur smell.

As HEYMAN1104 said if this was a transmission issue you wouldn't smell this only in reverse as the only item that would cause a smell with a manual transmission would be the clutch. If it was slipping you would smell it much more often and you would be able to feel it slipping as the engine would rev up without an increase in speed to match.

P0446 is usually a wiring fault to the vent control solenoid that opens/closes to let the fuel system equalize. It could cause a fuel smell but usually not a different smell.

A catalytic converter doesn't have to be ridiculously expensive. Buy it yourself online and take the car to just about any mechanic and they'll pop it in there cheap. I do but think it has to be welded, but even if it does you should still be able to find a smaller shop to do the install. All told, you can probably get out for $200.00 to $250.00. Which is enough, but cheap for a catalytic converter.

Whatever you do, don't buy a universal converter. Aftermarket is fine, as long as it's not universal. You'll spend more money buying pipe and having a muffler shop so the work than you would save to begin with.