I'm at the part where Manny is at the dam of bones in the Petrified forest. I can get the bone from the dam, throw it in the river and take out the extinguisher, but when the beaver leaps off I can't make him trigger the extinguisher. The icon does not display anything to indicate that it's possible, I just get a white cross, and no matter where I seem to put the cursor it doesn't change. After the beaver has come and gone, I can use the extinguisher just fine.

Is it a timing issue, is there something I'm not doing right, or is it something else?

The mouse controls seem pretty iffy in general, though I haven't even tried them in this part.

Maybe they don't work properly for this part or you're doing something wrong, either way, just use the original keyboard controls in this case. (Enter or U to use the fire extinguisher.)

Edit: Or maybe you're just getting the timing wrong. You say you're trying to use it when the beaver leaps off, but you actually have to start spraying a little earlier. That might be it.

Yes, this would be my guess too.
I made it through the entire Year One part using only mouse controls, and the flaming beavers part was not difficult at all. But to be fair, I have played the game a gazillion times and I knew what I had to do. The mouse controls did not make that section any more difficult. I got it first try.
This was with the original version. I don't know if the latest patch ruined anything with mouse controls.

The mouse controls are perfectly fine. I did this myself the other day without problem.

The trick is to do it asap, do not wait around for them to jump. Take out the fire extinguisher and fire it immediately.