Pretty much your answer right above but you're doing a lot of extra steps. Very simply, you could transfer the beer into the keg, chill it down, attach gas at serving temp, and leave it alone for a week like you are already doing.
Leaving it at 20 for a week is too long. Usually when people are doing these rush carbonation methods, they move on to drinking it that next day or so.
If you are wanting your beer quick, you could search on Amazon for carbonation lids. I have a few of those. They give step instructions but I just leave gas attached at serving temp for 24 hours then if a space is open, throw it in the kegerator. If not, leave it in my backup fridge. I've left the gas hooked up for more than 24 hours before without issue. Guess it reached the limit at that pressure and temp. Happy beer once it gets on tap.
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