Can GFP reporting be used to track localization of peptides in the ER, Golgi, and plasma membrane?
Historically speaking, the use of VSVG (Vesicular Stomatitis Virus G protein) labeled with GFP was indeed a practical method to define the secretory pathway. (@tyersome comment)

However, I am concerned that by the time everything is set up, the cell will be at steady state, which means GFP signal would be present throughout the secretory pathway, preventing me from doing this tracking.

Sec mutations in S.cerevisiae: Many biochemical pathways are initially identified owing to a particular mutation in the synthesis process. Recall Beadle and Tatum experiment and the logic they used to clarify the existence of 3 enzymes used in arginine metabolism.

(Image taken From https://www.chegg.com/homework-help/questions-and-answers/following-experiments-beadle-tatum-searching-mutations-disrupt-arginine-biosynthesis-pathw-q41797559)
A similar approach was taken back then to identify the distinct parts of the secretory pathway. Five different mutations in S.cerevisiae were identified named from A to E. Culturing all A to E mutated strains plus VSVG-GFP assay would clearly demonstrate the entire route.

(Image taken from https://slideplayer.com/slide/4382105/)
For example, even if the mutated class B, reaches the steady/equilibrated state, transport vesicles, Golgi, and the rest will not show the fluorescence activity (characteristic of VSVG-GFP) because all of the proteins have accumulated in the previous step(i.e budding from Rough ER)
Further Readings :
https://pubmed.ncbi.nlm.nih.gov/2188733/
https://www.sciencedirect.com/science/article/abs/pii/0092867481900647
MCB Lodish et al 8th edition chapter 14 section 14.1
https://www.nature.com/articles/35073068