I used this Gam3Gear Brook Switch Pro/Xbox One/PS3/PS4/PS5 to PS Classic/PS2 Super Converter With Gam3Gear Keychain https://www.amazon.co.uk/dp/B075462YRF/ref=cm_sw_r_apan_i_9RKZWW33JASMZAS1MGD4
You can power it with the ps2's usb (i did this with the fat model port at the front. It says it supports wireless though i had trouble with that. Works beautifully with ps4 controllers wired.
I bought 2 for multiplayer. One of my (2) usb ports were taken up by the power cable for my hdmi adapter, but i found i could power the second gam3gear brook by plugging it in to one of the ports on the other one.
Bear in mind ps2 era's game mechanics make FPSs a little more clunky than nostalgia tinted memory will remember, I still managed to complete tsfp on everything other than co op with these
You could try an upscaler. A little more expensive than just the cord, but it seems to be the most used option for retro consoles on modern tvs.
Most people seem to recommend this.