https://screenrant.com/batwoman-ruby...ails-evidence/
So the Baxter guy isn't credited with any association with the production, and according to Rose he isn't listed on the call sheets she has. The crew also aren't allowed to talk to the press about the production because they are under NDAs.
According to the timeline, the allegations about Batwoman being the last show still shooting add up, as Melissa Benoist (Supergirl) was known to be doing ADR from her home at the time.
Screen Rant Timeline:
May 17, 2020 - Reports Indicate Batwoman Barely Finished Shooting New Finale Ahead Of Government Shutdown
The COVID-19 pandemic led to widespread delays across the entertainment industry, from summer movies and theater closings to production on television series. While most of the Arrowverse series were forced to cut their 2019-2020 seasons short because of the COVID-19 pandemic, Batwoman was the last to stop filming according to Ruby Rose. This seems to be confirmed by one report regarding the rush to finish the finales of Supergirl and Batwoman from the day the final episodes of Supergirl season 5 and Batwoman season 1 aired. Reportedly, at the same time Melissa Benoist was recording ADR in her closet via Skype, Batwoman was still shooting new scenes on set. Again, this seems to confirm Ruby Rose's allegations that Batwoman continued filming until the Canadian government was forced to shut them down.May 18, 2020 - Batwoman Showrunner Confirms Cut Supergirl Scene
One day after Batwoman's season 1 finale, showrunner Caroline Dries spoke about how the planned Supergirl Batwoman season 1 cameo was cancelled due to COVID-19 filming restrictions. According to Dries, the two Arrowverse superheroes were supposed to appear in a photo together, with the frame concealing the Kryptonite shard Supergirl gave Batwoman during Crisis on Infinite Earths, but there was no way to get Ruby Rose and Melissa Benoist together to take the photo. This seems to contradict earlier reports that Benoist would be playing Supergirl in a Batwoman season 1 episode and begs the question of why they couldn't create a composite shot of the two actors using digital editing software if both shows were still in production at that time that the photo needed to be taken.