LM324 op-amp wheatstone bridge amplifier formula
Your problem resides in the oversimplification of $$V_{t+}$$

If you take directly the tension divider between the R and R(1+delta) resistors you get instead
$$V_{t+} = V_{REF}\frac{1+\delta}{2+\delta}$$

Which is approximately the same as what you used, but yields
 the good formula.

$$v_o = \frac{R_f}{R/2}\frac{\delta V_{REF}}{2(2+\delta)} \approx \frac{R_f}{R}\frac{\delta V_{REF}}{2}$$