Here is my perspective as a second year graduate student and recent TA for an introductory differential equations course (in Fall 2019):
Spend one lecture or less going through the second approach. While you should talk about linear functionals, most students won't have the necessary background and will therefore be confused and potentially get angry at you. This isn't your fault - the only sort of math they have seen has been the computational mathematics taught to them in the first three calculus courses. Some engineers like learning a lot about math (and some don't). If your class is mostly engineering students then they will be more concerned with the practical applications of the Laplace transform. At the end of the lecture, you could bring up a more technical remark. You could also invite your students to office hours and recommend additional reading if they want to learn more.
In terms of what to cover, I would make sure to explain the informal or non-rigorous definition of the dirac delta "function" provided $f(t)$ is continuous on an open interval containing $t=0$
0, & t\neq 0 \\\
\infty, & t=0
with the appropriate graph (when I was a TA, the professor provided a hand waving argument which showed that the function would have infinite height and an area of $1$ beneath the curve. None of the students objected to the hand waving argument).
The second property I would cover is
which can be visualized by choosing two or three example functions for $f(x)$. I would then use the definition of the Laplace transform to show the shifting property
which is true provided $t\ne a$ and $f(t)$ is continuous on an interval containing $t=a$. I would then show that for $a\ge 0$ this implies that
which leads to your example
in which you could go through the linearity of the Laplace operator (if you haven't done so in a previous lecture) and the fact that you will need to use
to find the inverse Laplace transform (assuming that students will be required to use the linearity of the Laplace operator and then find its inverse).
Outside of this, I would include anything about the delta "function" which you think is important. You could go through a second example similar to the one above where a different technique is used to find the inverse Laplace transform (such as a partial fraction decomposition followed by a different problem which can be solved directly through formulas previously derived). You could also spend more time graphing the other two properties to show how the delta "function" interacts with different example functions of $f(x)$. If you have $40$ or $50$ minutes to lecture, then you could spend the last $10$ minutes talking about what a distribution is (even though students wouldn't be tested on this material - some students will certainly be interested in learning more).