@zennistrad "Refurbished" by who??? I wonder how thorough?
Amber light indicates charging and flashing red is either the battery itself or the charging circuit is defective.
During "With the battery inside it will boot briefly but then immediately shut down" does the fan come on at all? Change speed?
If the fan speeds up you could try checking the heat paste on the CPU - too much, too little or too old paste could cause heat problems.
Some battery torouble shooting:
https://pc-dl.panasonic.co.jp/itn/manual/cfc2/P-2012-0313-B/referenc/manual/en/chapter15.html
Tear down:
https://www.youtube.com/watch?v=p4AuiaLjx8g
Try unplugging hard drive, WiFi card, etc. to eliminate a defective part as cause of problem.