Ok, so following the advice from @hellomacos I did the following.
* I tried using ifix it tweezers: they were too thick and so a no go
* I then looked for some precision tweezers on Amazon, and found these (Precision Tweezers Set, XOOL 10 PCS ESD Tweezers Set) https://a.co/d/6KvOvAA
* I used the thinnest tweezers in the set.
* I tried grabbing every way I could and what ended up working was to come from the top with the tweezer arms vertical, grab as far down on the SD as I could. Then lever the tweezers arms so they were placed more horizontal.
* Then held the tweezers as far down as I could to have maximum closing strength since the card was stuck in the locked in position.
* I then pulled and wiggled a little (no idea if wiggling helped) and the SD card came out!
* I finished by cleaning the SD port from the SD card shreds and done!
I finished putting the cover back (WITHOUT THE SD CARD) on and checked that the new SD card was locking and reading correctly in the Steam Deck (it should show up in the settings to format the SD card)
Thanks for yalls attention, and I hope this doesn't happen to anyone else.
TLDR: ALLAYS REMOVE SD CARD BEFORE OPENING STEAM DECK. Use precision tweezers to pull SD card, clean port, and check mechanism and port functionality.