Dead Battery(?) BIOS Update help

Which Linux distro are you using?
Ubuntu 22.04.3

Which kernel are you using?
6.8.0-52-generic

Which BIOS version are you using?
3.06

Which Framework Laptop 13 model are you using?
11th Gen Intel

Backstory:
About a week ago, my 3 year old Framework 13 laptop went off whilst hooked up to a USB-C hub on my desk that was charging it. It was connected to the top left side USB-C and the power indicator was blinking green on and off.

I did some reading through support articles etc. and basically concluded I should pop the CMOS battery and the main battery to reset the mainboard, so I did this (Guide used: /Guide/Fully+Resetting+the+Mainboard+State/113). It worked briefly but then went out again shortly afterward. The guide also suggested that if this problem re-occurs my CMOS battery might be done. I didn’t really want to reach out to support and wait a bunch so I found a supplier just outside of Toronto that I could drive to and get a replacement (this is my work laptop and if it dies my company will make me use a Macbook).

I swapped out the CMOS battery, but I have now come to the conclusion that my main battery is actually the cooked one. It appears to be only retaining a 1wh of it’s 55wh capacity (at least that’s what the battery measurement reported from upower). I read that there may be a bug where if the capacity of the battery gets sufficiently low the laptop won’t boot even if it’s connected to AC.

The temporary workaround I found to get back on my feet was to just disconnect my main battery (and it remains disconnected). This has worked, but obviously I am now hamstrung because my laptop is now in effect a desktop.

I couldn’t order a replacement battery directly from Framework because both the 55wh and the 61wh capacity are out of stock in Canada. I found a third party supplier that has 61wh framework batteries, but I noted when looking at the 61wh on the framework website that I need to be on a newer BIOS version than my relatively ancient 3.06.

So now I’m in between a rock and a hard place, because I cannot do a BIOS update without a battery (or I can but all the articles make doing a BIOS update without a battery sound particularly dangerous: Getting started guide with a bare motherboard), but the only battery I will have in hand is one that if it’s connected to an out of date BIOS will result in “Spicy Pillow” whatever that means (Framework 11th gen bios battery compatibility).

I also note that my crazy out of date BIOS may be a contributing factor to the failure of my battery capacity from reading the changelogs + updates so I don’t know if it’s the case that a BIOS update may be sufficient to get my 55wh battery to a state where it at least doesn’t cause my laptop to shutdown when it’s connected to AC power.

Question:
Is it better to do the BIOS update without a battery and risk it, or risk connecting the 61Wh battery and quickly do the BIOS update once the battery is connected?

Edit: Am also open to none-of-the-above suggestions if there are other solutions

For anyone who finds this in the future, I ended up reconnecting my near-dead battery, leaving it to charge overnight (powered off) and then doing the BIOS updates via an EFI USB stick.