In its original sense, 遅刻する refers to a punctual act of arriving late. So, if you are talking about a late arrival that is likely to happen, you should use 遅刻する.
仕事初日に遅刻するだけじゃなく、〜
Not only I’m going to be/arrive late on my first day on the job, …
However, 遅刻している is also fine as it is understood as referring to either one of the following two states.
仕事初日に遅刻しているだけじゃなく、〜
a) Not only I’m (running) late for work on my first day on the job, …
b) Not only I’m making the mistake of being/arriving late on my first day on the job, …
The first is about the fact of being late, whereas the second sounds like an answer to the self-accusatory question of What am I doing?! I know these two are different, but I’m find it hard to explain...
When said about yourself, 迷子になっている or 迷子になってしまっている would be understood similarly to (b) above. It’s good as a statement about an ongoing state of another person.
迷子になった or 迷子になってしまった refers to the event of getting lost that has already happened and thus put you in the current state.