I stick with the too fat spokes, and just keep them tensioned right. That's pretty snug, but never over tight. It's looser than a thin spoke wheels tension for sure. I check for over loose spokes perhaps every 500 miles or so. Which turns out about 4 times a year. To check, I just squeeze on each single crossed pair of spokes, looking for them to all feel the same. Plucked, they ring, but not a high C note. Nor a bass note.
But a new wheel needs attention every 20-50 miles, during the first 100. It depends on how cheap the too fat spokes are. The cheap ones will stretch out for awhile. Better spokes, it's more like tighten up once and done, even for the too fat ones.
Anything talking, like click click click every turn, then I stop and check, and tighten that loose spoke immediately, carrying a spoke wrench on the street rides.
Like AW,, if it really needs truing bad, likely the rim got bent by a pothole or whatever. Running a wheel like that is a disaster brewing. You have one side spokes all loose, the other side tensioned to the breaking point. Then a good bump breaks all those over tight spokes. One nice thing about disk brakes, you can still run a pretty damn wobbly wheel, while waiting for the new rim and spokes to arrive.