Nice watch Jabs. Difficult to give a name for this one since case date is not consistent with the movement date.But I would go for 6513. Let me tell some back ground on the case and on the movement.
The case is given the 12 number from around 1923. It has the very nice triangular flower side engraving, 5 in total on each side (count teh points of the triangles), that in later models increased to 9. You see that also the sides of the bottom plate are engraved. You see that only in early cases with 12 and 13 numbers. The bezel engraving has this curly pattern and not the expected laurel leave patterns that so many other models have. So far, I have seen this only on this model with case numbers starting with 12 and 13, like yours, but also on some early Princines.
The movement is from 1924. It must be one of the last 6TAA's since this one is the only one I have ever seen with a date code. Normally they come without date code and are from 1923, but always signed with "Bulova Watch Co." so from after April 1923. In 1924 the 6TAA got succeeded by the 6TA that all have a date code, either Asterisk (1924), either circle (1925). The earliest 6TA I saw has date code Asterisk with number 31362, while your 6TAA is number 27024. So indeed yours must have been one of the last 6TAA's before they changed it to 6TA.