History of the pronunciation of 辰 Today, it is pronounced chén in Mandarin. 

However, in Chữ Nôm, this character is used for the sound thì. This is surprising, since most Chữ Nôm characters either borrow the meaning of existing Chinese characters, or the pronunciation of a particular Chinese word. 

In chunom.org, it is indicated that 辰 was pronounced like “zhin” during the Tang Dynasty. This doesn’t really correspond with the Vietnamese pronunciation. The meaning of the character doesn’t correspond too. 辰 generally has to do with timekeeping in Chinese, but in Chữ Nôm 辰 is used to write the connector “so, then”. 

Were there any developments in the phonology of this particular word during the transition from Old Chinese to Middle Chinese, which resulted in such a divergence?  Or is this scenario a random error which cannot be explained using Chinese phonetics?
There is no contradiction. The formal Sino-Vietnamese readings of 辰 are thần [tʰən˨˩] and thìn [tʰin˨˩], which correspond nicely to the Middle Chinese [d͡ʑin] (Pulleyblank-Miyake), Old Chinese [djɨn] (Baxter). Compare 臣 with the same MC reading, also having Sino-Vietnamese thần. The Fifth Earthly Branch is called thìn (or thần) in Vietnamese.
However, the reading thì definitely exists. The confusion stems from the fact that it is a Nom reading, not some formal Sino-Vietnamese. Its type is either C2 (using a character to spell a local word with a similar sound): I believe that both thì 'time' and thì 'topic particle' are attested as 辰. thìn 'preserve' and thời 'time' are also occurring spelt 辰, and with the variety of Nom, it shouldn't be surprising.
Source: Tự Điển Chữ Nôm Dẫn Giải