Hmm, I don't think we have enough information yet to figure out exactly what's going on, so let me ask a few follow-up questions and suggest some possible causes.
Does your network dataset have a TimeZone table and attribute? If not, then I don't think the problem you're experiencing has anything to do with the date line because the network wouldn't even be aware of it.
What projection or spatial reference are you using? I believe some projected coordinate systems break in the middle of the Pacific Ocean and actually don't wrap all the way around the world. So maybe you need to choose a different coordinate system.
Have you verified that the network is actually connected in these locations? You can use the Explore Network tool to click on the line and see whether it's properly connected to its neighboring network edges. If it's not connected, you might need to read about network dataset connectivity: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/understanding-connectivity.htm
Are you solving your Service Area or Route using a time of day, or is it time neutral? If you solve it without setting a time of day, do you experience the same problem?
Maybe a dumb question, but did you remember to build your network dataset after making any edits to it?
(Please also tell us what version of Pro you're using.)