Your topic title says “simple”. Abandon all hope of simple.
I researched this topic heavily when I was looking to create a thermally balanced chamber for my P1P. The P1P lends itself to many more options because of the open frame nature. Note that the X1C has the same frame as the P1P but then you have to get over the mental hurdle of discarding one of the “pretty” side panels in lieu of a chamber modification if you go the route of an external venting system with a heater supplement. I ended up doing neither because it wasn’t needed when I found the right filament. But I digress.
Filament
While I digress. You mentioned you having problems with PC but it doesn’t sound like you’ve exhausted the filament option. I am a big fan of PC for precision parts. It isn’t really useful for making artistic model but for a filament that can be machined or tapped, there is nothing else. The filament I found that works well for me is marketed under various brands CC3D, HZST3D and now SHINA on Amazon. The seller is sketchy has hell in that they change names every few months indicating that they are hiding from bad reviews. Their PLA sucks but their PC is best in class when it comes to performance, albeit, not pretty.
You have to look for spools with this diamond pattern to know that they came from the same factory.
Passive Heating
On the subject of heating. You have two ways you can go, you can do passive heating by insulating the interior. That works and I’ve used bubble wrap insulator for that purpose. The silver bubble wrap works well because it insulates for both radiant and convective heat. Alternatively, I’ve used the bubble wrap envelopes that build plates come in. They are the perfect size so don’t throw them away.
https://www.amazon.com/s?k=thermal+bubble+wrap
Active Heating
Or you can go active which you cited above. There are hundreds of solutions that are internal to the chamber that will do a better job than the car heater you showed above. These are purpose build for the task.
https://www.amazon.com/s?k=3d+printer+enclosure+heater
I’ve not found a small space heater that is wirelessly controlled and if you think about it, the use-case for such an item is so narrow, there really isn’t a market.
Thermostatic control wired and wireless.
However, that’s where thermostatically controlled outlets will serve the purpose.
https://www.amazon.com/s?k=wifi+outlet+thermostat
X1E upgrade
And last but not least. You could go through the effort of retrofitting the X1E chamber heater to the X1C. I hate Bambu’s approach. The only advantage is that it utilizes the chamber control logic. Their use of blower fans is one that you will quickly hate. If you have no experience with blower fans, they blow air 90 degrees. Not surprisingly so does your hair dryer and that’s exactly how loud a blower fan can get at full speed so unless you love noise, stick with a larger fan.
Convective heat plate
You mentioned putting a heater in the bottom of the chamber. I hadn’t though of that but that could be made to work and possibly work well. Most filament dryers and food desiccators use these pads as part of their system. It wouldn’t take much to fabricate a blower fan to circulate air underneath the pad and just ensure that air is moving internal to the chamber. Either way, it would only cost you a few bucks to find out.
https://www.amazon.com/s?k=reptile+heat+pad
And of course, they can be controlled via a cell phone.
https://www.amazon.com/dp/B08Y5JDYFY/