We don't have that level of tech in the events of the Shepard Trilogy.
While that could work, you'll still have to deal with all the other problems involving the core discharge, like finding a place to do it in the perfect spot in intergalactic space 101,520 times.
What?
No, they can't. They have to organize the funds, draw the schematics, invent technology that breaks the laws of physics as they understand them, test said new technology, build the ships, and gather all the resources and personnel they are taking. And do all of this in a way that nobody finds out, which means stretching the project out to hide the trail and preventing any leaks(something they couldn't even do with the uber top secret Crucible, which people knew about within a few months) so that nobody else knows that the project happened hence why it isn't mentioned in the previous games. That is impossible to do in that timeframe.
There's nothing simple about this project. That's the problem. The level of technology and understanding of the universe our cycle has is not capable of this trip in the time frame allotted.
No ship in our cycle can do it.
But this thread is about James Vega. There are other threads where thinking Ark Theory works even though it can't without retconning the franchise or a DEM can be discussed.
I'll make it short since you apparently know very Little about physics. Yes, it can be done, it's current tech, we can use heatsinks that radiate the heat and surplus energy into the vacuum of space, you don't need any special or perfect spots in space to accomplish a cool down.
If the crew is in stasis then it doesn't matter how many stops your Autopilot does or how many hundred or thousand years the journey takes.
Stefan-Boltzmann Law
The thermal energy radiated by a blackbody radiator per second per unit area is proportional to the fourth power of the absolute temperature
You can use space heatsinks that radiates the energy.
In short, They have to organize the funds, draw the schematics. Which is a far easier Project than the Crusible in ME3, and far less space magicy since the tech exists today.
Heat and even electrical charges turn into radiation Waves that leves a hotbody and cools it down. Sicne it isn't new science it won't requier anythign too fancy. Having a detachable crew pod is nothign new, the moonlanders detached the landing module, in this case you would only need to detach it temporarily at Close range possibly with one shieldedmirror 0.00001mm thick if you want to stop pretty much all heat from possibly transfering(doubt anythign like that would be needed but it could help)..
Proper heatsink design would speed up the vacuum discharge, and disconnecting the habitation module from the drivecore module would make excessive heatbuildup a lot more manageble during discharge.
How hard do you seriously think that would be? You don't need any new tech, you don't need anythign that breaks science as they know it, you just need a few engineers and perhaps a few scientists to hammer otu a new ship design using known science nad Technologies. Then allocate the Resources to build it.
After Sovereigns attack on the Citadel in ME1 they would have been expected to get down to work on it. The Coulncil knew abotu the reapers. They didn't doubt it, but they wanted to avoid panic and did what they could to do what they thought was best for the preservation of civilisation. In ME1, they edknoledge it, and the Citadel DLC shows that Spectres and up were allowed to know about the Reapers. Meaning Spectres, Councilors or anyone working on a Project regarding the Reapers or an Ark Project.
The tech needed wasn't unknown, but it wasn't worthwhile or practical if you got relays and planets on which to discharge faster than you could in space.
As for build time, the Crusible was built in far less tiem while the galaxy was getting torn to bit's and pieces by the Reapers, compared to a golden age of economic sucesses and progress across Council space in between Me1-Me3.