I don't think you would have any problems regarding available power with nearly any power supply. The general recomendation of 400 W is to power a more than average PC. Remember it is supposed to power a CPU (about the same or sometimes more power than the GPU itself) and lots of low power parts like memory, HDD, motherboard, peripherals, USB ports, fans and sometimes diverse LED lighting. In our case the power supply must only power the GPU and eGPU adapter, wich probably doesn't draw more than a few Watts. I´d guess 20, 30 W tops, considering what it does and specially the absence of a cooling system (fans or fins) in a package this size and covered in plastic.
Considering we are hardware enthusiasts I think you would hardly buy a really cheap bad quality power supply for that project. It could not be good to your brand new GPU because of bad "quality" power (voltage spikes, fluctuations, switching noise...). So in case you decide to do it the right way, nearly any average to good quality power source would be able to very comfortably power any external GPU. As long as you respect the power drawn in the GPU power conector you should be fine.