Press the T key (for Talk) and then enter THEREISNOTRY. That's the command to skip to the next level. There's also PINOTNOIR # to skip to a specified level, though that doesn't work with the unofficial patch.
BTW, as an alternative to skipping the entire level, you could simply enter JEDIWANNABE 1 (I believe ON also works) to enable invincibility mode, either to play through the whole level like that or just get past the part that you're having trouble with.
That's a DFII code.
I can't get the old Dark Forces 1 codes to work either. Not sure if it's the glovepie and dosbox mappings interfering or something else.
On a default installation of the GOG version, typing the LASKIP code works for me (it must be entered during gameplay, unpaused). If your key-mappings are changed, you may need to take that into account. You'll have to determine which keys are mapped to which letters, and make sure you type them in the right order. Probably the best way to do this in-game is to create a new agent profile, and when you have to enter a name, see which key produces which letter.