If you think the tile is plugged and the tile is in good shape, I would run a sewer tape(flat steel tape about 100' long) You could run from your side and / or your neighbors breather(if he does not care).

If you find a blockage, mark the tape, pull it out and lay it on top of the ground, to mark the blockage. Assuming you can follow/site  the underground tile by probing.

We have found dead raccoons, tight as a cork.  Breather outlet, is it screened for varmit entrance?

Also, since you say it is shallow, a clay top could have collapsed, and once that is removed/fixed, it may clean itself out.

This way, you could dig a small spot( instead of 250') to fix the blockage. And your neighbor should not object to that.

As some people here like to say.... Post back with your experience.

Brian