Don't cheap out on your power supply. If something goes wrong, it can permanently damage the other components in your system, thereby costing you a lot more than $200 you would have spent on the PSU. If you need to shave money off the cost of the project, choose a less expensive GPU or CPU.
The power supply I have for my own luggable build is an AthenaPower AP-U1ATX40P8, a 400W ATX power supply in a form factor for 1U servers. Costs $130 new on Ebay:
https://www.ebay.com/itm/165816960542
AthenaPower has a few others too, but they are obviously going to be more expensive the more power you need. I will warn you that they are pretty loud, given that they need a lot of airflow for the small space they occupy. The fan speed is dependent on power consumption, so you can avoid some of the noise by using lower power components or by using software controls to restrict power usage.
Another option is to get a power supply of any size, and a few mountable ATX cable extenders to turn your PSU into an external power brick. It may be a pain to have to unplug the 24-pin and plug it back in every time you want to use it, so you could get a PicoPSU in addition to your regular PSU and send power to the PicoPSU from one of the 12V connectors. Crypto miners mining CPU mineable coins do this a lot in order to allow a larger power supply to share power with multiple motherboards.