I can't find any help with Google on how to install custom firmware on a PSP 3001 with v4.21 firmware. I'd prefer to do it without the Pandora battery, but if that's the only way then can you please describe how to do it in a way that's safe for this version?
The current de facto standard custom firmware is PRO-C2, which supports the PSP-3000. There is no need to use the pandora battery, recovery mode, or other exploits, and it can be (and indeed should be) installed over the latest official firmware version.
The official install instructions are on the PRO CFW wiki, but the short form is:
Note that there's no support for permanently patching the bootloader (so that you don't need to run FastRecovery) on the PSP-3000, but everything else works fine. And this also means that if you want to use the plain Sony firmware, you just reboot the PSP and don't run FastRecovery and there you go.
(Disclaimer: I don't actually have hands on experience with the PSP-3000, but this is how it works on the -1000 and -2000, and the developers say it works on the -3000 as well.)