Hi, got the same problem. :(
Now I got two shield tablets, one with a really bad bad battery and one completely new and NVidia also told me to dispose of the "broken" one.
I found this while searching for an adequate battery:
That guy opens it up and also shows a comparison between original and replacement.
And further research told me that getting the "original" battery is impossible but a replacement "as close as possible" is very well, possible.
The replacement I found is a little "weaker" in capacity and also a little bit smaller (by mm's) but still better than a "soon to explode" one.
It goes by the name "LG L238" or sometimes "Lenovo L238" (150x64x3,5mm 5000 mAh) - 3564150.
I found three resellers on aliexpress... I don't know their credibility... anyway here are the links:
Keep in mind, you will have to solder the original batteries loading circuitry to the replacement battery.
I haven't done this myself yet because I'm hesitant to purchase from aliexpress since I have NOOO experience with them.
I'm still looking around if I can find a local supplier with acceptable prices (not 50€+).
Hope I could shine a little light at the end of the dark tunnel... maybe ifixit has more knowledge for a fitting replacement battery and suppliers. :(