It's been broken for a long time. You used to be able to turn off 2FA so you could sign into both the PS3 and PSP with your regular account password. But that no longer works and PS3, PSP, and Vita will only allow you to sign in with a device password.
You can setup a legacy access point, I've set on up with a Raspberry Pi. That will let the PSP and original DS get online. And the PSP can still download games from your account download list.
If you mod your PSP, there is a plugin that adds WPA AES support so the PSP can connect to modern APs. You still need to enable legacy login on the AP as it doesn't support the very latest tech. But it's much better than running an old network. You can then again access your download list.
Or if your PSP is modded, just install whatever you want on it. Unless you are hacking online games Sony hasn't banned anyone for modding a PSP or Vita and playing games even when getting trophies or playing online matches.