Here you are working over $\mathbb{C}$ (or perhaps any other splitting field of characteristic 0 for $G$). So the representation you are starting with is just the direct sum over all characters $\chi$ of $T^F$ of the various induced characters from $B^F$ to $G^F$ obtained by lifting $\chi$ first to a character of $B^F$ and then inducing. All of these induced characters of $G^F$ have the same degree, but some are irreducible and others not (as in the extreme case $\chi =1$). So the resulting endomorphism algebra of the large direct sum will be cumbersome to study. It's helpful to consult Chapter 10 of Roger Carter's 1985 book for a more precisely organized program along these lines, due largely to Howlett and Lehrer. Naturally the usual Hecke algebra for $\chi=1$ plays a role here, as do analogous endomorphism algebras for other $\chi$.

But your expressed hope seems too loosely formulated in this extremely complicated situation. Have you tried to work this out explicitly when $G^F = \mathrm{SL}(2,p)$? In that case all the induced representations are easily identified.

By the way, there is a version of all this worked out in the defining characteristic $p$ by Carter and Lusztig in their old paper *Modular representations of finite groups of Lie type*, Proc. London Math. Soc. (3) 32 (1976), no. 2, 347–384. They use BN-pairs as a framework and develop intertwining operators in the spirit of Hecke algebras, but with some degeneracy.