Some suggestions.
I rest all my "sporting" rifles at the entry pipe. If I had a heavy bench gun that did not have a full or 1/2 stock with rod underneath,
I would sure rest it out on the barrel, testing location for accuracy. The further out you can rest it, the more movement it takes at
the rear end, to move the sights.
I have always, with 1/2 stocked & full stocked rifles, achieve the best accuracy at the entry pipe on the sand bag that is atop the adjustable rest.
The rest's elevation and windage adjustments are locked, so there is no vertical shaft movement in the rest itself. Any movement
in itself will add to group size.
Adding talc to both front and rear bags can also reduce group size.
I have not tried one of those Caldwell bags, that have a long V in them, approx. 18" so the forend is held quite well, and there is no need for a rear bag.
I see some of the air rifle testers using these. They look interesting, but as yet, I have not tried one.