Approval before I modify the form factor of an ATX PSU
There should be no problem if you:


Make sure you don't short anything out.
Ensure enough cooling via the new fans to compensate for the one you are removing.
That the new fans don't leave a pocket of air or under cooled part because of a change in direction/blocked path.


Easier solutions would be to either get a smaller power supply, mount it only partially in, or mount it externally and feed the wires through.

There are a ton of different power supply form factors. ATX supply only refers to the power output and connectors for a standard ATX compatible pc/motherboard/peripherals. Some are shorter, longer, wider, thinner, all four sides smaller, weird L shapes, etc.