This is very dependent on what you want from this rifle. Is this an heirloom type piece? Is budget important? Is this something that the child will keep and use for the rest of their life? Is this the rifle that all future children of the family will learn to shoot with?
I think the most...
American Persimmon
Wild plum
Allegheny chinquapin
American hazelnut
Blackberry
Without removing some larger trees I think your main issue will be finding something that tolerates shade.
3-9x is surely dated and probably compared to modern scopes only useful for certain hunting applications.
A big reason, in addition to weight and cost, is probably increased light transmission with the relatively large objective lens on 3-9x, usually 40mm+ compared to 24mm on most LPV.
I am considering a Griffin Optimus Micro for my 2nd .22 can.
It can be configured with several different mount options and can handle 22lr through 22-250 with some barrel length/mount restrictions.
https://www.griffinarmament.com/optimus-micro-silencer-22lr-through-22-250/
With "breaking the bank" left undefined. This will put you in the middle. Not $$$ but not cheap.
https://www.precisionfirearms.com/m8/6.5%20Grendel%20Criterion%20Barrel--6-5-grendel-criterion-barrel.html
plus...
The iron sight are the back up to the red dot, and the red dot is zeroed?
I would just verify the BUIS are co-witnessed to the red dot and confirm the BUIS zero next time I got a chance.