$L^{\infty}$ as colimit
The answer to this question is YES -- but it is useless!
In fact, a theorem of Valdivia (which you can find, e.g., as Theorem 6.5.8 in the book Barrelled Locally Convex Spaces of Bonet and Perez-Carreras) states that given any infinite-dimensional separable Banach space $F$, one can represent any ultrabornological locally convex space $E$ (this means, that $E$ has some representation as an inductive limit of Banach spaces) which does not carry the finest locally convex topology as an inductive limit of Banach spaces isomorphic to $F$ such that, in addition,the linking maps between the steps are nuclear operators.
This theorem is rather useless because it is too good: Every ultrabornological space has such a representation and thus you cannot deduce any nice properties from it.
When I read your question, I assumed that you were asking whether there is a way to do this in the context of the underlying structure rather in the spirit of the answer above which addresses the situation where the spectrum consists of spaces which are merely isomorphic to the appropriate ones. Here are some suggestions (rather than an answer).
For clarity, I will work in the context of a space $\Omega$ with a $\sigma$–algebra $\cal A$ (rather than the reals) and the $L^1$-case. The family of positive finite measures forms a lattice and so the corresponding family $\{L^1(\mu)\}$ can be regarded both as an inductive and a projective spectrum in a natural way. So we can take inductive and projective limits in various senses.
Your query refers to the former but I am afraid that it seems to me that the results are not of interest (a purely subjective opinion, of course). One can take projective limits in three senses—that of Banach spaces (with linear contractions as morphisms), strict topologies, and locally convex spaces. Only the first two are of interest in the context of your query.
The first case yields the Banach space of bounded, measurable functions, the second one the same space with a weaker, but still complete, lc topology. If you are interested in duality theory, I would suggest that the latter space is the interesting one—its dual is the space of $\sigma$-additive measures on $\cal A$.
It's a very short answer but this may be of interest to you:
Davis, Henry W.; Murray, F. J.; Weber, J. K. Jr., Families of $L_p$-spaces with inductive and projective topologies, Pac. J. Math. 34, 619-638 (1970). ZBL0187.05802.
https://projecteuclid.org/download/pdf_1/euclid.pjm/1102971942