I tried the Ruger sp101 .22 yesterday at the range. While I liked the gun in SA the DA trigger pull was very heavy. I found myself just shooting SA the whole day because of it
First off welcome to TFL.
2nd,,,yeah,,,,that's a very common thing.
Sadly, all too common.
I fell into that trap once upon a time.
I bought an H&R Convertible (.22lr/.22mag) D/A to try to polish up the D/A skills.
That thing has a horrible heavy D/A trigger.
Thankfully, it resembles a S/A more than it does a D/A.
Bad D/A pull leads to frustration, which leads to shooting a D/A revolver S/A.
Since the idea is to sharpen up the defensive skills, all it does is burn ammo and build bad habits, since in a defensive situation you'll never use S/A on a D/A revolver.
Personally, I'd just as soon jump right into a good .38/.357 mag, get into reloading, and work on the D/A trigger that way.
That's how I learned and it worked out well for me.
An alternate would be a S/A .22 such as you mention since the gun is different, both in function and in feel.
Or another option would be a good .22 semi auto.
Both single six and a good semi auto are well within the price range you listed.
Used S&W D/A revolvers, such as the old Model 17 and 18 are in that ballpark also - but - they don't come easy. You'll have to really search for them.