How should I run power to a powered bidet toilet seat?
You're not allowed to use extension cords as a substitute for the fixed wiring of a structure.   NEC 400.6 -&gt; 400.10 in the new Code.
What you want is surface conduit, something like Legrand Wiremold, starting at an existing recep or switch, and running along the wall or baseboard to where you need the power.   There, you install a receptacle box compatible with that conduit system.
The receptacle or switch where you started the conduit, gets moved outward onto the starter box's surface.  From there, you run THHN individual wires (black white green), to the new receptacle.  It's a lot like building LEGO but with some hacksawing on the conduit segments.  If you don't want to do the arcy sparky stuff, you can install the physical conduit (that's most of the work) then call a pro to connect the wires.
Further, you'll really, really want this extension circuit to be GFCI-protected, and the best way to do that is tap an existing GFCI receptacle.