I have not read the whole thread, but here is my 2 cents worth.
i would guess that bil and fiance did not remember/know the date of the lw's anniversary, even if it was his brother's wedding. Not everyone remembers dates of others anniversaries. I would also guess that this date worked for them in terms of the time period they wanted to get married, when the venues were available, etc. Also, to them it may not be a big deal that their anniversary is the same as the lw's.
It would be interesting to know if the ILs said anything to bil when he anounced the date, if they remembered the date of their son's wedding the year before and mentioned it, if they think it isn't a big deal, if they think it is cool that both sons are married on the same date.
LW and her husband should go ahead and celebrate their 1st anniversary, which is special, as they want to celebrate it, whether that is dinner out or a weekend trip. If it works to go to the engagement party, fine; if not, that is okay. I can't imagine anyone getting mad that they are celebrating their first anniversary. Now, they do need to go to the wedding and participate in pre-wedding events that they are able to. They can celebrate the 2nd anniversary another time, as many of us do--we don't all celebrate on the exact date. However, I do think the first anniversary is a little more special and if it is important for them to celebrate that day, they should do so.
I sincerely hope that this isn't a plot by bil to hurt his brother in any way by having his wedding on the same day and then taking that day as hostage--celebrate our day, celebrate us--for the rest of their lives. That could make things very uncomfortable down the line.