So, I finally got the Vidmaster Endure achievement. For those who were having the same problem I was, it may very well be the people you are playing with. Through playing other firefight maps for the 200k achievements, I narrowed down who has the connection that drops at random times. Turns out though, that it was two of the guys who were in the original Endure party; this is why it seemed like it was a game issue. So after trying Endure with both of them, and then once with each of them by themselves, I tried it with neither of them and we successfully made it to the 5th set sans lag out.
One thing of note though, one of the guys who kept causing the lag out had a wireless connection while the rest of us had wired. Not sure if the other guy did too or not but if you're having the same issue then try to have everyone wire directly to the router.
Good luck.