The brass can be expanded to the proper diameter after sizing.
How that is done on a Whack-A-Mole setup, I do not know. My press has an expander die that is pushed into the case mouth.
One could hone out the neck portion of the sizing die, making the size larger. This is a non-reversible change, that should be done carefully to avoid damaging the die. There are die sets that use bushings to control the neck diameter, as well. These use a press though.
The "Brit" die is of the right neck diameter, but I dont know how it could be made to work.
Perhaps a cone shaped bit of steel could be tapped into the case mouth to flare it enough to shove a bullet in. I would be wary of going to far and causing chambering problems, but a little might get the bullet started.
A boat tailed bullet would be easier to get started too, though I know you have some to use already.
Try the three zero eight sized bullets first. They might be fine. Saving a lot of work and worry.