Here are two other approaches to the formula, one of which may satisfy you, depending on your tastes:
Roquette's Analytic Theory of Elliptic Functions over Local Fields includes a self-contained discussion of this formula in Section 3 on pages 23 to 29. Roquette's approach, though, is (non-Archimedean) analytic and also uses a bit of the theory of genus-1 function fields--not really low-level.
In the paper where he introduces the tau function, "On certain arithmetical functions," Ramanujan sketches a direct proof of something related to $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$, namely $\wp'' = 6\wp^2 + \frac{a_4}{2}$ (Equation (21) in that paper). One can find some further details of the computations in Chapter 4 ("Eisenstein Series") of Berndt's Number Theory in the Spirit of Ramanujan, but Berndt leaves part of the work as an exercise for his readers. A more comprehensive treatment, generalizing Ramanujan's approach, is in the first chapter of Development of Elliptic Functions According to Ramanujan by Venkatachaliengar and Cooper. The authors reach $(\wp')^2 = 4\wp^3 + a_4\wp + a_6$ at the bottom of page 11. Perhaps by reducing the generality of Venkatachaliengar and Cooper's intermediate formulas, one could produce a minimal low-level proof that fills in the details of Ramanujan's sketch.