Unreliable SMS/MMS delivery between AT&T and Republic Wireless

I have a 1st gen Moto X on AT&T, and my girlfriend has a different Motorola phone (don't remember the model off the top of my head, but I think it's one of the budget ones) on Republic Wireless. SMS and MMS messages are often delayed or even dropped. This has been happening on and off for around a year now, but recently has increased in severity. Here are possibly relevant details that come to mind:

• I am running Android 5.1, she is running 4.4. We are both using the stock messaging app and are not using Pushbullet/Airdroid/similar.
• She doesn't have a data plan, I do.
• Although Republic Wireless routes SMS/MMS through Wi-Fi when possible, the problem appears whether she's on Wi-Fi or not.
• Delay times started at 3-5 minutes, but recently have approached 45 minutes or even an hour. The frequency of delays has also increased.
• Only one SMS has been fully dropped, as opposed to the vast majority of MMS.
• MMS that have been delivered have caused her phone's messaging app to crash when she opens them. I haven't done any real troubleshooting on this, instead, I've just stopped sending MMS. I think this is an issue with her phone rather than either carrier, as the crashes stopped when she deleted our conversation.
• Recently, her phone's messaging app will occasionally crash when trying to send SMS to me. No other contacts with this issue yet, but she doesn't text that much. I will ask her to text other people and see if she can't get it to happen again.
• I don't have any issues sending SMS to anyone else, regardless of carrier. I haven't ever sent MMS to anyone not on AT&T, but family members who have report similar delays/drops.
• Rebooting either phone changes nothing.
• There is no discernable pattern to when delays happen other than the recent increase in frequency.

I 'm admittedly not that familiar with cell networks in general, but I'd be inclined to guess that something is getting lost in translation between the two networks. However, I don't have any concrete evidence as to what is actually going wrong.

Are there any obvious troubleshooting steps that I've missed? Any other instances of this problem? I've done some preliminary googling, but haven't turned up anything useful.