I didn't think Mei was so bad here. She did communicate. She made it clear that Arata is not and should not be considered a rival for her affections whatsoever. She reaffirmed that they're engaged and strongly questioned why Yuzu would even consider anyone a rival still.
If anyone is being the same, it's Yuzu. Mei realizes Yuzu is uncomfortable and says as much, but she also explains to Yuzu that she has no choice but to work with Arata to complete this important project. Yuzu shouldn't be so worried after that, yet she still is even though Mei has never shown a single drop of interest in Arata. If anyone is having to do work for the relationship right now it's Mei not Yuzu.
Then we see that even in private Mei is worrying and thinking of how Yuzu is feeling. If anyone is being unreasonable, again I'd say it's Yuzu. She should have more trust in her partner and should recognize her issues as personal ones she needs to work through. It takes two to party and Mei has made it clear Arata isn't even an option, she only sees Yuzu. So what he feels is irrelevant, exactly what Mei said. This is literally Mei's job and Yuzu is throwing a fit because of a coworker based on gasp Mei having conversations with him! The horror. I hope they don't confront Mei in her office over something she hasn't even done. It's these things that make Mei upset and I can't blame her. Work is stressful enough.
In the beginning, I think Yuzu became insecure because of how quickly Arata changed his attitude when Mei found him yelling at Yuzu, and when he realized it was Mei standing there in the Manga it appeared as though he was going to hug her when she brushes him off because she didn't recognize him.
When Yuzu wakes from her weird dream about feeling lonely. The way I would interpret Yuzu's dream is that she was feeling left out and lonely because once Arata realized that the girl who came to Yuzu's rescue was Mei he started trying to catch up. When Mei rescues Yuzu in front of the men's restroom to when Mei and Yuzu board the train going home Arata dominates Mei's time (taking Mei away from Yuzu) and doesn't ever include Yuzu in the conversation.
Yuzu learns a lot about Mei and Arata: that they were childhood friends, they were both the children of the headmasters, and they hadn't spoken in 8 years. All the things she learned just made her feel even more left out of the relationship between Mei and Arat.
Upon hearing about Yuzu's dream her friends turn the dream from Yuzu's feelings of loneliness into Arata now becoming a love rival. In the Manga, you find that he is setting it up so that he can spend more time with Mei through his father and Hemiko reveals that when he was younger he said that he wanted to marry Mei. So, I wouldn't say that he doesn't still have feelings for Mei but I will say that Mei has assured Yuzu that she doesn't have any feelings for him. Mei lets Yuzu know that she is engaged to her and there is no need to worry about Arata, she will only be communicating with him as part of her work.
When she was telling her friends about her weird dream of being a dog and feeling left out because everyone else was a bear (more specifically, Mei and Arata). Her friends made it seem as though the dream meant that she was insecure about Mei and Arata's relationship and that he could possibly be a love rival.