Oh wow, so a lot has gone on.
So, do you think that her boyfriend was the one affecting her behavior? I mean, no offense to your ex, but in our communications over the past year, I got the impression that she wasn't...I don't know --emotionally stable? Like she could be all over the place, and definitely didn't act with your daughter's best interest. Was that all him?
What is your lawyer suggesting then? That you drop things for now? I'd agree it's a heck of a lot harder to make a case when a situation is stable (because why change what isn't broken, right?), but does that just undo everything that happened up to this point? And who's to say that you drop things and then she reverts back to her old ways?
Ultimately, I know you want what is best for your daughter. You always have. If your ex stays nice, you guys get along, you follow the orders laid out in your agreement, and so forth, do you think that would be sufficient for you? I can't speak to your reconciliation - I don't know if that is something you even want or not, but I wouldn't be swayed by the wants of her mom in making any such decision. Perhaps the best thing, if that is something that BOTH of you are open to, is to go to some marital counseling first, while still apart, and seeing if you can work through your issues.
As for the argument, yes, it's unfortunate, and stuff happens. I know you were only looking out for your daughter, and arguably, your ex could have just said "No, he's not here, just my friend is over, the television is on loud in the background is all." If you want to do right, even though you're not entirely at fault - call and apologize. Say what you did to me - that you made a mistake in thinking the television was her ex, that your daughter had expressed to you being uncomfortable in the household with him recently, and you only meant to protect her. You never meant for it to lead to a heated argument, and if things were said, you regret it.