Question about thevenin circuit analysis with voltage across voltage source
If your objective is to find the Thevenin resistance, then I'm not sure why you're approaching it this way.

If you can find the Thevenin resistance \$R'_{th}\$ of the circuit to the left of the \$1\Omega\$ resistor, the total equivalent resistance is just

$$R_{th} = R'_{th}|| 1\Omega $$

And, I recommend using a current test source which will make finding \$R'_{th}\$ almost trivial.