Maybe a lace-like see-through pattern instead of a full cutout could lessen the chance of paper jamming.

Hmm, yeah. An option.

I don’t know what the intent of the see-through is, so this idea may have no value for your specific situation this time — but what about just perforating the logo and letting the readers interact by punching it out themselves?

The first page of the body has an image that we wanted to reveal through the cover. I’ll make a note of that, Kent, but I’m not sure if it would work for this.

Just fake it: print the front cover with what looks like the next page showing through a hole.

Or, depending on the size of the print run, have the die cuts done manually after the piece is bound.

I've used both both die cuts and translucent paper on jobs I've designed, but always with saddle-stitched binding, not glued. Your printer would probably be the best source of technical advice IF they have previous experience with these techniques on glued covers. Can they show you examples of similar pieces they've done? If so, then they would probably be the best source of advice. But if not, then perhaps you should be looking for a different printer for this job.

From a design standpoint, my concern about translucent paper is that the entire page would show through, not just the area you'd planned to die cut. Unless maybe you have large inked areas to cover what's on page 2.

Either use translucent paper or follow Nick Shinn’s suggestion and fake it. Even if the cover comes out of binding okay, you still have to contend with automated mail processing equipment—unless you go to the additional expense of poly-bagging the magazine before mailing.

A “screen” of multiple mini-perforations might prove more robust, and yet still transparent enough to make the effect.

Glue the die-cut cover to an uncut sheet of clear plastic?

You could use a loose dust-cover, eg not glued to the book. Fold over the first and page of the book, with these being a thicker weight of paper, maybe.
This may involve some handiwork (do it by hand instead of automated), but is a guarantee that you achieve the desired effect.

Used to be that esp French books were sold this way (promotional dustcover on simple book block).

Thanks guys. All good ideas. If I come up with a viable (automated and not too expensive, since the print run is rather big) solution I’ll let you know.

Perhaps the solution is just to look for an alternate design that doesn't involve real or simulated die cutting.

The 21st century equivalent of die-cutting is laser cutting.

The publisher I work for has done this for several books -- rather small print runs, though. But much to my surprise, this could even be done after a book was perfect bound. In that particular case it had to be done after binding, because it had those colored inside pages that get glued to the cover (uh, I cannot recall the English term for this) and the idea was to see the colored page through the cover, rather than the original white "page 1".

Theunis, that sounds fascinating. I didn't realize that was possible. Do you have links to any photos of those books?

James, quite annoyingly I only have a link to its e-pub version. We did ask, but laser cutting iPads was not recommended by its manufacturer ...

To get the idea, though:

This is the front cover design. The front was printed in black only; the text "insight" and the red Wolverine slashes were cut out to show the red of the inner cover pages.

Lucky you weren't laser cutting a word with any O's.

We did for another job!!

But we used a Stencil-like font for that title ;-)

those colored inside pages that get glued to the cover (uh, I cannot recall the English term for this)

Theunis — I believe you are referring to what we call the “end papers” in English.