When dealing with combinations of materials, it is better to deal with modulus weighted section properties (EA, EI, GJ) rather then simple area section properties (A, I, J). Assuming the outer shell is perfectly bonded to the inner foam, the stiffnesses of each will add to get the total stiffness, Think of combining springs in parallel. Therefore, compute EA of the foam, EA of the shell, and add them to get the EA of the composite. Similarly for EI and GJ.

If you really need to have the A,I,J separately (say to model the whole as a beam in FEA), you can chose a reference moduli, call them Eref and Gref, and compute the "effective" section properties by dividing: Aeff = (EA)total/Eref, Ieff = (EI)total/Eref, Jeff = (GJ)total/Gref. That allows you to define the "area" of the beam and the "modulus" of the beam as separate values as required by many programs.

Another comment: I am guessing that the stiffness of the foam is much less than the stiffness of the carbon fiber shell. In that case, the main thing that the foam will probably do is to stabilize the stiff outer shell. That is, is prevents the shell walls from local buckling under compression (or torsion) and it prevents the circular cross-section from ovalizing under bending. In that sense, this is similar to a flat sandwich panel, where you have strong/stiff outer skins with a weak/soft inner core.

Yes, ignore the foam, analyze as a simple round tube. For axial stiffness and bending stiffness calcs, use Ex moduli (assuming x is in the length direction of the tube). For torsional stiffness, use Gxy in the calcs.