Yeah, I believe that may currently be the fastest card available with models that don't require external power. If the PSU were not a concern, it doesn't make much sense at this point, seeing as a 1650 SUPER typically costs just a little more for around 30% more performance, but that card also draws more power. The 1650 (non-super) performs behind the RX 570 as well, but that card can draw more than twice as much power under load. A 1650 will still offer multiple times the performance of what you have, of course, and should be capable of running just about any current game at 1080p, provided you are willing to lower settings in demanding titles.
As for the power requirements of a 1650, I suspect you should be able to get away with that 220 watt PSU, assuming the motherboard can supply enough power through the PCIe slot. A 1650 only draws around 70 watts under load, and the other components in your system probably won't push total power draw from the PSU to over 200 watts while gaming. Just make sure it's a version of the card without a power connector though, as some require them. And of course, make sure it physically fits in your case, if it's a small case.
I do suspect the PSU is probably replaceable though, otherwise it would be difficult for the manufacturer to do warranty repairs on the system. Are you saying it's hidden behind a metal shroud? What case is this, or model of PC if you don't know the specific case? Perhaps a photo of the "irremovable metal plate" could be helpful. Have you checked if the right-panel on the opposite side of the case might be removable to access the PSU?