SUM of values in a comma delimited cell

I'm not sure I understand the need for B1 string...

If as advised you create Names _A, _B, _C etc... (you shouldn't / can't use C for ex.) such that _A refers to =$E$1, _B to $F$1 etc... then you can use what you already have ie:

A1: _A,_B,_C,_D
B1: =Eval

... the only limitation to the x,y,z rather than x+y+z is that the former is constrained in terms of how many items you can include in the delimited list but I suspect that won't be an issue here, ie:

=SUM(1,2,3)

is the same as

=1+2+3

(edit: in fact the SUM route has added advantage over + given the lack of explicit coercion taking place - ie non-numeric names won't generate #VALUE!)