Since you have plenty of water, can you raise the dam?
I like the way you think, but that's not really possible.
Also, ponds have a limit called "carrying capacity". Most simply, there is a limit on how many pounds of fish a given pond can support. If you have lots of big golden shiners, they may actually be using up some of the pond resources that otherwise would go to the BG.
For a small pond like yours, you might be able to actually fish out enough of the shiners to make an improvement.
They're tough to catch with hook and line, but easy to cast net. I haven't tried to systematically remove them but I do catch a bucket full from time to time to use for bait. I'll also clip the tails and release (a tip I saw here!). Seems to work as I haven't caught one with a missing tail.
3'-4' IS shallow and you will be constantly battling weed problems. The pond may get too hot for the HSB, but I don't think it'd hurt to throw 15-20 in there and see if they make it. You probably won't see them for the first year or two if they do make it.
There really just don't seem to be any weeds at all. Not saying they won't appear at some point. If I get the chance, I'll grab some of those HSB and report back how they work out.
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Speaking of carrying capacity, the pond was full of massive bullheads when I moved in and I pulled probably 50 of them in the 2+lb range. Very few intermediate size. I thought I was done with them (got really sick of eating old catfish), but last night I put out some lines and got two 3lb'ers and a 2lber. Maybe I should have tried to make it a trophy bullhead pond?
