There is no such thing as a CJP flare bevel or flare V-groove if one stays with prequalified joint details.
The weld size is a function of the corner radius, which for the purpose of calculations is a function of the wall thickness. If D1.1 is the governing document, Table 2.1 lists the weld size for the prequalified grooves in question. The weld size can be increased if the WPS is qualified by testing per clause 4.11.5.
Regarding your original question is this post, the figure C-4.1 may provide a little insight. According to the figure, the longitudinal weld is considered to be a fillet. If that causes your head to spin you could use the correction factors of Annex B to calculate the required weld size needed to produce the weld with the necessary effective throat. Most designers ignore the subtleties.
Al