Do you think Eckels understands the seriousness of the situation? Do you think Travis is overreacting?
The premise of the story "A Sound of Thunder" is that any change to the past, regardless of how small, can result in tragic consequences in the future. A ripple effect of changes would be set off if the past became different, and just as ripples in a lake get bigger as they spread from their spot of origin, these changes would get bigger as they got farther into the future. This is the accepted position on time travel.
Therefore, "Time Travel" and all of its employees, including Travis, must make sure that under no circumstance do their clients fail to understand this.
Eckels displays a very cavalier attitude from the beginning. He never seems to really take it seriously until it is too late.
In response to the question and in light of all of the above:
No, Eckels never really seems to have understood.
It is hard to accuse Travis of over reaction based on the extreme consequences of Eckels' actions, but there is probably room for debate over the final events of the story.