Is there a way to reflect small edge-weight changes after computing Floyd-Warshall on a large graph?
Yes.  The phrase to search for in the literature is "dynamic shortest paths".  "Dynamic" refers to the path that the graph might change, and we want to efficiently update the shortest paths (more efficiently than recomputing them from scratch).

There are different ways one might change the graph.  Some papers consider the case where edges can be added.  Others consider where edges can be removed.  You can also find algorithms that can handle both kinds of changes; that might go under the name "fully dynamic shortest paths".

Search the literature.  You'll find many algorithms, with different tradeoffs between running time, simplicity of implementation, and types of graph updates supported.