In the past I have put the computer on it side and had a PSU sitting on the frame next to the DVD drive so the cables would fit. It's not something you walk away from but for a quick power on test it did work
Exactly what I do all time, I just hook the known good PS to the MB and see what happens, if the PS is sitting on the bench right next to the MB I cant see any MB being so BIG that the cables wouldn't reach. The same if you are onsite just plonk it next to the computer connect the cables and power up.
Why on earth would you take out the old PS put another one in and screw it all down connect the cables. Just plonk the PS next to the computer, connect, fire it up and see what happens. Once you've confirmed the PS is bad then order the new one with longer cables.
Its the same if you had a problem with a DVD drive, first thing I do is put a commercially produced DVD in the drive, if that doesn't work connect a known good DVD drive and see if it works, if it does then install the new DVD drive, why would you take out the old one, put the new one in screw it all in ? its madness.... just plonk it next to the machine on the bench plug it in power it up and away you go.
Maybe I am different, I come from a background in electronics where we did component level repairs, nowadays that is too expensive, so much more efficient and cheaper to just swap out the faulty part, eg I would never bother to repair a faulty PS, I just replace it with a new one.
I know from my 30+ odd years of doing electronic repairs that the PS doesn't need to be screwed into the case to work. Even if for some weird reason it needed for the case of the PS needed to be electrically connect to the computer case you would just use a patch cord with alligator clips on each end and connect it that way.
rgds
Syb