Installing CFW on 6.20

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I have a PSP 2006 without a battery and it won't allow me to update to 6.61 OFW without it. Unfortunately it's too much of a hassle to acquire a battery where I live, and it's not really worth it for what is a retro console
I was wondering if I can just do it on my current firmware and how the compatibility with the games that were released for the newer firmwares is?
Or if there's a way to update to 6.61 without a battery, I'd really appreciate it
Actually, I do have a battery but it's really swollen and barely fits inside the console
What are my options?

submitted by /u/phantasmagore48
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