In general, I don't think a rolling block is the right action for a benchrest .22.  However, I had a liner for mine that shoots right up there with a dedicated match stevens 44 I also own.   I somehow misplaced my liner, might have left it at the range.  I need to make another. 

One problem with the rolling block, as you discern is that it doesn't readily seat a bullet into a match chamber.  Your thumb gets tired of pushing them in.   

The hammer fall is also a bit more energetic than a .22 needs.  That jars the rifle.  It's hard to get the trigger under about 12 ounces and still be safe.

Other than that, any of the rolling blocks will work.  I would use a full size #1 for a bench rest gun.  A #2 would also work, but a #4 might be a bit light.

dave

Guys, the title says we're looking at a suggestion for a .22 rimfire rifle...the hammer fall is going to have more force than the recoil with a No. 1 or No. 5 action!  In this situation, I think that I would look at the No. 2 action...more beef than the little No. 4 and lighter than the No. 1.  Even though it only came with plain triggers, a good smith can work it down to a moderately light pull...if you insist on going with the Remington action.  Personally, I have not felt at a disadvantage with an underlever action on the bench.  Possibly if I had a rifle with a large, ornate, Schuetzen lever I might, but with a standard lever, it usually a non-problem.   

David

John, if you raise the bore line to put the firing pin at 6 o'clock, and make an extractor that goes past the chamber to the right, it would give a bit more support to the rim on firing.  This might work best with one of the sliding bar extractor versions...

David

Well I dug out the action and all the parts I could find. Looks like I need a breach block. May as well make one from scratch and put the firing pin off center for the rim fire. I thought I had an old breach block with the ear missing that I could weld up and use but it was not in the parts box. I have some heavy walled tube and a bunch of 22LR rifle barrels so the plan is to take a 16" barrel and press it in to a longer tube of about 30" and octagon it. This will give a free bore of 14" but will give the rifle the feel of a large bore ( with very little recoil). 
Thanks for the idea  Irascible.

John,

I'll send you a breech block with no pin hole drilled.  It saves a lot of the work over making one.  Wait a few days to start the project, I'll have it in the mail to you monday.

regards,
dave

Interesting, I have read many articles which either said 19" or 20", but I never saw 16" mentioned. What ammo?

Well the barrel is ready for threading. It's 28" long, about a #3.5 tapered octagon. Has 16" of rifled barrel and the rest is free bore at 5/8". As soon as the breach block is here and finished the barrel will be threaded and head spaced.