Is it wrong to break the bread after the Sacramental prayer is offered, rather than doing it before?
No, there isn’t anything wrong per se. But that is the pattern we have used in the Church. And there is a rhyme and reason for it.
Imagine that you’re sitting in your pew and we’ve just had the blessing on the bread. Then you’re sitting there waiting for the 2 to 4 minutes it takes to break the bread. This eats into the sacrament meeting time that usually ends up going over as it is.
Well, we want to have a sacrament hymn to prepare us mentally and spiritually to renew our covenants. And it just so happens that most sacrament hymns take about as long as it would take to break the bread. So, it’s a win-win.
In the New Testament the Lord did it in the reverse sequence.
22 ¶And as they did eat, Jesus took bread, and blessed, and brake it, and gave to them, and said, Take, eat: this is my body.
30 And it came to pass, as he sat at meat with them, he took bread, and blessed it, and brake, and gave to them.
Looks like he blessed it first. Then break.
My understanding is that in times past, this was the method. And this was partially because that is how people socially broke bread (literally). Not only that, but it was only broken in half and handed out to either side of the one offering. Then each individual used to break off a small piece and handed it down further. This is obviously too unsanitary for our culture to consider. Hence they have the priests use sani-wipes at the sacrament table.
Then in the Book of Mormon we read:
And when the disciples had come with bread and wine, he took of the bread and brake and blessed it; and he gave unto the disciples and commanded that they should eat.
So apparently it has been done both ways.
You may also consider how you dish yourself up at the dinner table. Do you dish up the plates first and then bless? Or do you bless then dish up? For my household it is different for each situation. We do what is most practical.
In other words, the sequencing is more about practicality than specific performance of an ordinance. However, this is the currently prescribed method. So, unless your bishop agrees that there is a good reason to go around it, for now, just go with it.