Would probably use CloudWatch Logs for this personally
It's probably cheaper but lambdas and s3 are equally dirt cheap so hey why not.
Edit: Two possibilities here. Either the choice of technology is deliberate and there's a reason I can't quite figure out, or AWS has reached that state where they do so much it's really really hard for people to know what service to use for each problem.
Which actually makes me wonder what am I doing right now that could be done easily using a pre-baked AWS solution?
For debugging situations, I'd like to send a request to my app server and then see the server response live, or with little delay.
awslogs get <group_name> -w
The latency is pretty low and this generally works well. You can apply filters etc. as well if needed.