History of the pronunciation of 辰
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