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](https://www.mathi.uni-heidelberg.de/~roquette/1970-AnalyticTheory.pdf)_ 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](http://ramanujan.sirinudi.org/Volumes/published/ram18.pdf)," 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](https://bookstore.ams.org/stml-34/)_, 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](https://www.worldscientific.com/doi/suppl/10.1142/8252/suppl_file/8252_chap01.pdf) of _[Development of Elliptic Functions According to Ramanujan](https://doi.org/10.1142/8252)_ 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.