You have a point, she is a bind - but I don't think Harima thought throughly about it, he might regret it later - because in my opinion Harima never thought about Tenma as rival but that it was another failure to impress her - so this time he is using Karasuma to impress her . Moreover, Harima might think that if he appears in front of Tenma as someone superior or equal to Karasuma, Tenma will take more interest in him.
In Ch. 217, Tenma told Harima that she would be competing against Harima Hario from here on. If Karasuma and Harima are coauthors using that same name, then to compete against "Harima Hario" means to compete against Harima and Karasuma. This places Tenma in a double bind situation, where she has to choose between competing against "Harima Hario" and keeping her relationship with Karasuma intact. Harima wasn't bragging to Tenma - he was placing her into a situation where her goals conflict with each other. Expressed differently, he is saying "Which do you choose - the guy or the manga?" similar to how Eri posed a similar question in Ch.214.