What are the widely used notations for quickly referring to a single component of a multi-component object?
Papers I've seen that have this issue often present 1 as the initial definition and then also define 2 (or 3 with subscripts) as an additional definition that is used when necessary for clarity.

A common abuse of notation—common enough that I've seen it in multiple places, although it makes my skin crawl every time I see it—is to write something like $S = S(TS)$.

3 with superscripts is unacceptable because superscripts have other widely-understood standard meanings; however, it would be fine if the superscripts were changed to subscripts. 4 is never used outside of programming contexts.

I would also recommend avoiding the two-letter variable name $TS$ in mathematical notation, especially since you are also using $S$ as a separate symbol. Use single letters, resorting to Greek if necessary, for anything in the standard italic "variable" font. Special functions and classes of languages usually get straight font or boldface to distinguish them from surrounding text for this reason.