Link and
Archive to Newsweek article, I don't think it's the same one you screenshotted; if anyone can access the Financial Times article it seems like they first reported the story, that one might be worth reading but it's paywalled.
The Newsweek story at least said who within the admin was responsible, Richard Grenell, who was ambassador to Germany during Trump's first term (and apparently was lacking in diplomacy in that role). He's currently "envoy for special missions", and in that role was previously sent to California to oversee the admin's response to the LA fires, and Venezuela to do diplomacy over deportations. He met in person with Romanian foreign minister Emil Hurezeanu at a conference in Germany (it's not clear whether meeting with Hurezeanu was his sole or main priority there), and before that there was apparently a phone call involving "US officials" during which the topic was discussed. I'd imagine the main topic was the recent annulled election in Romania; the article may be slightly misleading in implying that these overtures were undertaken specifically because of Tate rather than him being one of several topics discussed, but it's absolutely clear from direct quotes that Grenell does support Tate and his incarceration is a priority, and he's not alone in this in the administration.
This is hugely embarrassing. At best the support Tate garners in right wing circles is a very unfortunate blindspot, but in this case it really makes me question Grenell. Oh yeah, and he's gay. Wait, maybe that explains it...