it rained a whole night and maybe even more in june, 1815 but that isn't exceptionel in Belgium.
In my lifetime I have seen hot weather in June, but also torrential rains in that same month in other years, as I have seen the same in July (we know our national day, the 21th of July, normally as the day with the drache nationale - or the national torrential rainday
And the same applies for August, or even for the whole summer.
So saying that a torrential rain in June was triggered by a volcano eruption is easy but no proof can be given as 200 years ago there wasn't any climatologic science nor annotations of weather systems. And even less proof can be given that this was the cause of the defeat of Napoleon at Mont-saint-Jean.
And even if it rained, it rained for bot parties, as also the Allies had to sustain that rain during the night

Best wishes

Marc