This idea won't work ...

a) The loads on the header from the branches are point loads, not uniform loads.

b) Even if you knew what the 3 forces and 3 moments were (at the intersecton point of each branch), imposing them on the header would only be correct for the initial evaluation of the system. As soon as the header moves (for any reason), these (imposed) forces and moments will be incorrect. What your overlooking here is that there is an interaction between the 1" branches and the 10" header. Granted the effect of the header on the branches will be more significant thant the effect of the branches on the header, but there is an interaction and you can't ignore it.