Hello,
As @2disbetter suggested you should reach out to Purism. If the battery replacement is free, why not? I had issues with my Librem13 V2 last year and they solved everything and were very nice about it.
As far as running the laptop without a battery, I cannot think of a reason why it wouldn’t work. But again, to be safe, you should reach out to Purism support: they’ve been great to me and should answer. 