Like others I am skeptical about the possibility of finding an elementary proof of the dimension formula which is detached completely from the character formula. But of course I can't prove an impossibility theorem. The history of Weyl's character formula shows that it can be approached from a number of different directions in creative ways, though it is hard to declare one proof "easier" than another since all of them presuppose serious background. Weyl's original method used integration on a compact Lie group (a convenient real form of a complex semisimple Lie group), exploiting the fact that every element has a conjugate in a fixed maximal torus. The proof highlights the importance of what is now called the Weyl group (but was used by E. Cartan), in its realization as the quotient of the normalizer of such a torus by the torus itself.
Once the character formula is in hand, a formal calculation yields the dimension as well. (It seems unlikely that the dimension formula alone could be strong enough to recover the full character, however.) There have been other approaches to the character formula, including a more algebraic but opaque one found in older textbooks. The approach of Bernstein-Gelfand-Gelfand in 1971 including their BGG resolution provided a much more elegant algebraic proof, pointing the way toward the more general treatment of infinite dimensional highest weight modules (Kazhdan-Lusztig Conjecture, etc.) and the version by Victor Kac for certain Kac-Moody algebras.
The Borel-Weil theorem using the geometry of line bundles on flag manifolds gave another interpretation of the character formula (as an Euler character), while Bott's further work clarified the behavior of sheaf cohomology for non-dominant weights. (But Bott's theorem isn't needed for the Weyl formula.) In the 1960s Demazure found a short, elegant proof of Borel-Weil-Bott in the setting of algebraic geometry and cohomology of line bundles on flag varieties; this proof uses only the most basic facts about algebraic groups and sheaf cohomology. Later Andersen and others Donkin showed how to streamline the derivation of Weyl's formula in this setting, while Andersen and others extended some of the ideas to prime characteristic: see Chapter II.5 in Jantzen's book Representations of Algebraic Groups. Offshoots of such thinking have permeated the study of Lie group representations, as well as representations of algebraic groups in prime characteristic or quantum groups at a root of unity.
While the Weyl character formula has been a catalyst for developments in representation theory of various flavors, the dimension formula is limited to the classical finite dimensional setting and seems to be mainly a byproduct of the character formula.