You are going about things the wrong way.
First:
TVout cards will never give you 240p via the component/svideo/composite video outputs.
The signal only comes out in an interlaced mode (typically 640x480i for maximum flexibility, as most PC VGA monitors at that time could simultaneously show that as 640x480p). They were designed mostly the workplace, where you might want to show a video, charts or a powerpoint presentation during training sessions and meetings. That kind-of-thing. Not really very good for, or intended for, gaming or as a second work screen.
Now, it is actually possible to hack these TVout cards, and force them into outputting
component @ 240p via the VGA head. I know because I did so nearly 20 years ago. It was a terribly arcane process. I had an ATI "Sapphire" 9650 (AGP slot!) bought new, a specialised dongle (VGA-Component) that came with the card, WinXP and some proprietary video mode software "Powerstrip" by Entech. The software was never continued past XP and Entech does not exist anymore, though there are probably old copies floating about.
Anyway, here is an abbreviated origin story. Back then (2006/7), this was the only way to do 240p component-out. Decent RGB-component transcoders cost ridiculous amounts, around US$250 (anyone remember Shinybow?), and this experience is why I thought to design/build GreenAntz transcoders. Because I believed nobody should have to go through the same crap I did to do arcade games on a component TV.
In other words, you don't need bother with TV-out or arcane hacking. GreenAntz is specifically designed to save you this hassle.
What I’m trying to do:
Ultimately, I’d like to play mame games on my CRT and utilize just two power outlets to do so (Basically, Comp -> GPU -> CRT). I mainly want to play games that MiSTer currently does not have a core for. I understand that this is a very specific use case scenario.
I don't understand the issue. You say "power outlets", do you really mean
video outputs? You want to be able to quickly and easily swap between PC and Mister?
Have you considered a KVM (Keyboard+video+mouse) VGA switch? KVMs are cheap and are great for quickly changing inputs between devices. You can have PC+ Mister, their own separate controls, outputting to just one transcoder and one CRT. There are different flavours, here is one:
https://www.amazon.com/Selector-Switcher-Sharing-Monitor-Keyboard/dp/B077QC7GNV?th=1I have a Radeon HD 5770/R9 280X that I’ve gone through the CRT Emudriver/Atom-15 process with and they work well (but don’t fit what I’m trying to accomplish) and I have a MiSTer (and am aware of MiSTer Cast, but again, doesn’t do what I’m hoping to do).
TBH, it seems pretty sweet so far.
What I have done:
I had a setup previously that was my Comp -> GPU -> Transcoder -> CRT. Although it worked, I never ended up liking that setup. I have one transcoder from the user Zebidee, one from Aliexpress that does pretty much the same thing and one from TVOne (1T-C2-400). The first two were very straight forward and worked but utilize a separate power source (Technically since they both have USB-C outs for power, I could connect them to a PC, but I’m unsure if that’s enough power or not). The TVOne works well but needs a level of configuration that can get pretty confusing but again, requires its own power source.
GreenAntz consumes very little power (around 80-85mA @ 5v, so about 0.4W), and works best when it shares the same power supply as the video source (minimises ground-level cross interference). Most commonly powered directly from PC via USB 2.0 to USB-C cable. GreenAntz does provide 5v and GND landing pads, but that is more for flexibility in arcade cab applications. If the KVM has a spare USB port (like the one linked), I imagine you could power it from there. With the KVM you only need one transcoder.
The Aliexpress one is similar as it is a ripoff of an old pre-GreenAntz design we were kicking around - does the job, but was something we thought was too crap to sell! Oh well

Attached a pic of a KVM switch - check your favourite online retailers.