The foreign travel shouldn’t be a big problem. If you look close there should be instructions saying you can add additional information on an attached paper.
They won’t necessarily contact your current employer but they will verify employment. Lots of ways for them to do that.
I’d say getting the position is probably a long shot though.
BTW, legal immigrants aren’t a problem.
1. A good freeper has no business trolling for these types of government jobs.
2. The department of the interior has no business running these types of extensive background checks. Another waste of money.