The H&R double action can't really be altered much.
They aren't a real good example of a D/A either.
In general though, a D/A rimfire is going to be heavier because it takes more of a strike to set off the rimfire.
On an H&R, you can grind off part of the hammer spring to lighten the pull a little.
Either have a gunsmith that's done it before do it or get an extra spring if you DIY.
I had a gunsmith do mine years ago and he got it down from "Godzilla" to about "Gorilla" level.
IIRC, he took off two coils - but - don't hold me to that.
The only other D/A rimfires I own are S&W M17's. The D/A on those is pretty good. Not as good as my M19, but, pretty good.
Wait - I forgot - I also own a Dan Wesson D/A .22. It's on sort of permanant loan to a friend of mine.
I think the D/A trigger on it stinks.
He thinks it's fantastic.
I own S&W
He owns Ruger.
That's probably why. The Dan D/A trigger feels more like a Ruger than a Smith.